Adhesive Joints – Theory (and

use of innovative joints)ERIK SERRANO STRUCTURAL MECHANICS, LUND UNIVERSITY

Wood and Timber – Why I’m intrigued

…to this!

…via this…

From this…

…and this…

Fibre deviation close to knots

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?

0.0001 m

10 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

Str

ess

Deformation

Str

ess

Deformation

Strength

Stiffness

Material models

• Less commonly used models

Str

ess

Deformation

Softening behaviour

Softening behaviourS

tress

Deformation

Strength

Fracture energy

Stiffness

Shape

Basics

• 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

E1, t1, b1

E2, t2, b2

G3, t3, b3

P1 P2

P3

P4

Luftfahrtforschung, Vol. 15, 1938

Basic variables

N1(L)N1(0)

N2(0) N2(L)

x

L

dx

N1 N1+dN1

N2 N2+dN2

t3

A1

A2

t3

Cross section

b

0

0

32

31

dxbdN

dxbdN

t

t

01

01

32

2

2

31

1

1

t

t

A

b

dx

dN

A

A

b

dx

dN

A

0

0

32

2

31

1

t

t

A

b

A

b

21, Axial stress in adherends1

Horizontal equilibrium

22

11

3123 /)(

u

u

tuu

Assumes constant shear strain

in adhesive layer

222

111

333

t

E

E

G

Assumes linear elastic materials

2a

3

Axial displacement of adherends

2b

2c

Derivation twice of and using and then

we obtain, using and :

2a 2b 2c

13

032

3 tt

22113

32 11

EAEAt

bG

with the definition of 4

5 Stiffness ratio shear/axial

Solution of governing equation

• The solution of is given by

where constants C1 and C2 are determined by the

boundary conditions

032

3 tt

)sinh()cosh( 21 xCxC t 6

Load case “pull-pull”

E1, t1, b1

E2, t2, b2

G3, t3, b3

P

P

113

32

22113

31

1

)sinh(

1

)tanh(

1

AEt

PGC

LAELAEt

PGC

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

Different y-axis scales!!!

L=100

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

• If

<< 1 (small) => uniform stress distribution

>> 1 (large) => non-uniforms stress distribution

22113

32 11

EAEAt

bG

L

L

Conclusions – so far …

• Parameter

<< 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

L

Joint load-bearing capacity

• Assume joint capacity is reached when max shear

stress equals bond line shear strength, then from

Where is the bond line shear strength

(Assumes is chosen such that for which case

max stress occurs at x=0)

6

1

22113

3max

)sinh(

1

)tanh(

1

LEALEAG

tP f

t

ft

7

111

22 EA

EA

11

22

EA

EA

Example: Influence of overlap length

Conclusions – so far …

• Joint capacity (N) depends on adhesive shear strength

and parameter

• Pmax thus depends on geometry, strength AND stiffness

parameters

1

22113

3max

)sinh(

1

)tanh(

1

LEALEAG

tP f

t

L

22113

32 11

EAEAt

bG

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 behaviourS

tress

Deformation

Strength

Fracture energy

Stiffness

Shape

Influence of rod length

Initial softening

Bonded-in rods

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

𝑏ℎ2/𝑓𝑚

Influence of joint brittleness ratio

No

rma

lise

d u

ltim

ate

str

en

gth

Joint brittleness ratio

Perfectly plastic

LEFM

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 layerNLFM

Comparison with tests – Pull-out

E. Serrano and P. J. Gustafsson. “Fracture mechanics in timber engineering – Strength

analyses of components and joints”. Materials and Structures (2006) 40:87–96.

FEM

Test

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

Gf

Material strength, ft

Displacement (mm)

Stress(MPa)

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.

Results – NLFM and LEFM

Compliance method (analytical)

Compliance method (analytical)

Lamination thickness (mm)

Bendin

g s

tre

ngth

(MP

a)

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

Brittle adhesive

Semi ductile

adhesive

Ductile adhesive

Shear stress

Peel stress

Brittle adhesive Ductile adhesive

Brittle adhesive Ductile adhesiveSemi-Ductile adhesive

DIC-measurements

Serrano, E., Enquist, B. Holzforschung,

Vol. 59, pp. 641–646, 2005

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

Softening behaviour S

tress

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?