0.00 0.01 0.02 0.03 0.04 0.05

0.00

0.01

0.02

0.03

0.04

0.05

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

r (mm)

z (m

m)

σyy/σmax

1.0

0.8

0.5

0.3

0.1

-0.1

-0.4

-0.6

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.02

0.04

0.06

0.08

0.10

r (mm)

z (m

m)

r (mm)

z (m

m)

σe/σy

1.40

1.26

1.11

0.97

0.82

0.68

0.53

0.39

0.24

0.10

Adnan Abdul-Baqi & Erik van der Giessen

Koiter Institute Delft

Delamination of a Strong Thin Film From a Ductile Substrate During Indentation Loading and Unloading

−1 0 1 2 3 4 5 6−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

∆n/δ

n

Tn/σ

max

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

∆t/δ

t

Tt/τ

max

(a) (b)

Introduction

Indentation loading and unloading of a strong elastic

film on a ductile substrate is modeled. The film is con-

sidered to be elastic, the substrate is elastic perfectly-

plastic and the indenter is spherical and rigid (Fig. 1).

The interface is modeled by means of a cohesive surface.

The cohesive surface model

In the description of the interface as a cohesive surface, a smalldis-

placement jump between the film and substrate is allowed, with nor-

mal and tangential components and respectively. The

corresponding tractions we adopt in this study are those used by Xu and

Needleman [1]. Figure 2 shows these tractions in their uncoupled form.

∆∆n ∆t

Figure 3. (a) Contour plot of the Von Mises effective stress at the maximum indentation depth( ), ( ). Arrows show the shear direction, line shows the location of the tan-gentially delaminated area. (b) Contour plot of the vertical stress component at the end of the unloadingstage ( ), ( ).

hmax 2t= φn 150= J m2⁄

F 0= φn 500= J m2⁄

Figure 2. (a) Normal traction versus normal separation. (b) Tangential traction versus tangential separation.

1 1.15 1.3 1.45 1.6 1.750

0.1

0.2

0.3

0.4

0.5

a

b

c

d e

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

h/t

F/(

πR2σ

y)

φn (J/m2)

a 150b 200c 400d 600e >> 600

Results

Figure 4. Load versus displacement curves for dif-ferent values of interfacial energies. Curve (e) repre-sents the case of a perfect interface.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

h/t

E/E

max

Eel

E

pl

Eint

∫ F dh E

el+ E

pl+E

int

Figure 5. Evolution of elastic, plastic, interfacial andapplied work during the loading and unloading process.

Conclusions

1. For relatively weak interfaces, delamination occurs in the loading stage. It is

imprinted on the load vs displacement curve by a ‘kink’.

2. Normal delamination occurs during the unloading stage, where a circular part

of the coating is lifted off from the substrate. It is imprinted on the load vs dis-

placement curve by a ‘hump’.

[1] X.-P. Xu, A. Needleman, Model. Simul. Mater. Sci. Eng. 1 (1993) 111.

(a) (b)

increasing

increasing

∆t

∆n

∆t 0=∆n 0=

h

R

a

t

interface

substrate

coating

h

Figure 1. Geometry.