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COHESIVE ZONE MODELS FOR INTERFACES

Piet Schreurs

Marco van den Bosch, Sil Bijker, Muge Erinc

08/06/11

3

OVERVIEW

With ongoing miniaturization, interfaces get more important con-

cerning the (mechanical) behavior of material systems. The degra-

dation of interfaces – e.g. delamination – can be studied and ana-

lyzed with cohesive zone models.

These models are used to investigate the behavior of interfaces

between polymer coating and steel sheet substrate, a material sys-

tem used more and more for all kind of applications. Also the

peel-off lid of food containers is relying on the proper delamina-

tion of a polymer interface layer. As a final example, the interfaces

in solder joints for microelectronic components are studied, with

special attention for their fatigue life.

The next sections are based on the research work of Marco van

den Bosch (PhD), Sil Bijker (MSc) and Muge Erinc (PhD), who’s

papers and reports are listed in the reference section.

4

OVERVIEW

1. Cohesive zone models

2. Exponential cohesive zone law

3. Coupling and mode mixity

4. CZ-element

5. CZ for large deformations

6. Application : polymer coated steel

7. Application : easy peel-off lid

8. Application : solder joint fatigue

5

COHESIVE ZONE MODELS

Cohesive zone (cz) models have been introduced by Dugdale and

Barenblatt and have recently attracted a growing interest in the

scientific community to describe failure processes and delamination

in particular. Cohesive zones project all damage mechanisms in

and around a crack tip on the interface, leading to a constitutive

relation, or cohesive zone law, between the traction and opening

displacement. Advantages of cz-models are: 1) interaction between

crack faces is automatically incorporated and 2) can be fitted on

experimental data.

Cohesive zone models relate the relative displacement (”open-

ing” ∆) of two associated points of the interface to the force per

unit of area (”traction” T ) needed for separation. Frequently – but

not necessarily – a difference is made between normal (n) and tan-

gential (t) direction, so the cohesive zone law comprises the two

relations Tn(∆n) and Tt(∆t).

Cohesive zone laws can be uncoupled or coupled. In an un-

coupled cohesive zone law the normal/tangential traction is inde-

pendent of the tangential/normal opening. In a coupled cohesive

zone law, both normal and tangential tractions depend on both

the normal and tangential opening displacement. Uncoupled laws

are intended to be used when the debonding process occurs under

one mode – normal (mode-I) or tangential (mode-II) loading – or is

largely dominated by one mode. The majority of cohesive zone laws

have a (partial) coupling between normal and tangential directions,

which is achieved by introducing coupling parameters in the model.

6

COHESIVE ZONE MODELS

t

n

traction = f(opening) T = f(∆)

normal / tangential Tn = fn(∆n) ; Tt = ft(∆t)

coupling Tn = fn(∆n, ∆t) ; Tt = ft(∆t, ∆n)

7

COHESIVE ZONE LAWS

A large variety of cohesive zone laws has been described in litera-

ture. Most of them can be categorized into the following groups:

(a) polynomial, (b) piece-wise linear, (c) exponential and (d) rigid-

linear. These four cohesive zone laws are depicted schematically

in the figure below, where in the upper row the normal traction is

given as a function of the normal opening Tn(∆n) and in the lower

row the tangential traction as a function of the tangential opening

Tt(∆t). The maximum normal traction and the maximum tangen-

tial traction are indicated by Tn,max and Tt,max, respectively, and

δn and δt are characteristic opening lengths for normal and tan-

gential direction. The areas below the curves represent the normal

and tangential work-of-separation φn and φt.

8

COHESIVE ZONE LAWS

α = {n, t}

(a) polynomial Tα = Tα,max

∆α

δα

f(λ) ; λ =

√

√

√

√

∑

α

(

∆α

δα

)2

(b) piece-wise linear Tt,max = (δn/δt)Tn,max

(c) exponential Tα = Tα,max

(

∆α

δα

)

exp

(

1 −∆α

δα

)

(d) rigid-linear Tα = Tα,max

(

1 −∆α

δα

)

9

EXPONENTIAL COHESIVE ZONE LAW

The exponential cohesive zone law is most popular. It has some ad-

vantages compared to other laws. First of all, a phenomenological

description of contact is automatically achieved in normal compres-

sion. Secondly, the tractions and their derivatives are continuous,

which is attractive from a computational point of view. The expo-

nential cohesive zone law originates from the universal relationship

between binding energies and atomic separation of interfaces.

The exponential cz-law described here, is based on a potential

φ, which is a function of both the normal and the tangential open-

ing. The potential incorporates four independent parameters: the

work of separation for pure normal opening, φ0 = φn, the work of

separation for pure tangential opening, φ0 = φt, the characteristic

opening in normal direction, δn, and the characteristic opening in

tangential direction, δt.

The normal and tangential tractions are calculated by differ-

entiating the potential w.r.t. the normal and tangential opening,

respectively.

For pure normal and tangential opening, the tractions are plotted

in the figure below. The characteristic lengths δn and δt can be

expressed in the maximum tractions.

10

EXPONENTIAL COHESIVE ZONE LAW

φ(∆n, ∆t) = φ0

[

1 −

(

1 +∆n

δn

)

exp

(

−∆n

δn

)

exp

(

−∆2

t

δ2t

)]

Tn =∂φ

∂∆n

∣

∣

∣

∣

φ0=φn

=φn

δn

(

∆n

δn

)

exp

(

−∆n

δn

)

exp

(

−∆2

t

δ2t

)

Tt =∂φ

∂∆t

∣

∣

∣

∣

φ0=φt

= 2φt

δt

(

∆t

δt

)(

1 +∆n

δn

)

exp

(

−∆2

t

δ2t

)

exp

(

−∆n

δn

)

δn =φn

Tn,max exp(1)and δt =

φt

Tt,max

√

12exp(1)

−1 0 1 2 3 4−1

−0.5

0

0.5

1

∆n / δn

T n / T n,

max

−3 −2 −1 0 1 2 3−1

−0.5

0

0.5

1

∆t / δt

T t / T t,m

ax11

COUPLING

Adequate coupling between the normal and tangential directions is

required in a cohesive zone law to describe the physically occur-

ring interface behavior realistically. If complete loss of interfacial

integrity is important (e.g. in the case of a moving delamination

front) and a cohesive zone completely fails in shear, its load-carrying

capacity in normal traction should completely vanish as well and vise

versa. To investigate this behavior, the interface is first loaded in

normal direction until a maximum opening ∆n,max. After that it is

broken in shear : ∆t → ∞, as is shown in the figure below. The

shaded areas under the curves are the work of separation for normal

and tangential opening.

12

COUPLING

Wn =

∆n,max∫

0

Tn(∆n)|∆t=0 d∆n

Wt =

∞∫

0

Tt(∆t)|∆n=∆n,max d∆t

• traction

• dissipated energy

• φn = 100 Jm−2 ; φt = 80 Jm−2

13

COUPLING : TRACTION

Several loading sequences are evaluated. In the left figure below the

evolution of the maximum shear traction is shown as a function of

the normal separation. As can be seen in this figure, the maximum

shear traction decreases to zero for increasing normal separations.

Theoretically, this can also happen in compression. However, these

negative values of ∆n cannot be reached in practice since the normal

compressive traction increases very fast in this regime.

In the right figure below the maximum normal traction as a

function of the tangential separation is shown. The maximum nor-

mal traction decreases to zero for increasing tangential separation.

14

COUPLING : TRACTION

T ∗t = max{Tt(∆t, ∆n,max)}

−1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

∆n,max / δn

T* t / T t,m

ax

T ∗n = max{Tn(∆n, ∆t,max)}

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

∆t,max / δt

T* n / T n,

max

15

COUPLING : DISSIPATED ENERGY

Coupling should also be realistic, when considering the total work-

of-separation Wtot = Wn + Wt in case of a sequential loading,

in normal and tangential direction or vice-versa.. The behavior is

as expected and shown in the figures below for values φn = 100

Jm−2 and φt = 80 Jm−2: in the case where there is first normal

separation until ∆n,max and then complete tangential separation,

Wtot increases monotonically from the value φt to the value of

φn. In the other case where there is first tangential separation

until ∆t,max and then complete normal separation, Wtot smoothly

decreases from the value of φn to φt.

16

COUPLING : DISSIPATED ENERGY

Wtot = Wn + Wt

0 2 4 6 80

20

40

60

80

100

∆n,max / δn

W [J

m−2

] Wn Wt Wtot

0 2 4 6 80

20

40

60

80

100

∆t,max / δt

W [J

m−2

] Wn Wt Wtot

17

MODE-MIXITY

The influence of mixed mode loading on the total work-of-separation

is studied next. To this purpose, a cohesive zone is loaded under

an angle α until complete separation occurs, and the corresponding

value of Wtot is calculated.

Wtot is shown as a function of the loading angle α. Values

φn = 100 Jm−2 and φt = 80 Jm−2 are used in these calculations.

In the figure below it can be seen that Wtot = φn when the cohe-

sive zone is only loaded in normal direction (mode-I: α = 0o) and

Wtot = φt when it is only loaded in tangential direction (mode-II:

α = 90o).

18

MODE-MIXITY

Wtot =

∞∫

0

Tn(∆n, ∆t)d∆n +

∞∫

0

Tt(∆n, ∆t) d∆t

0 20 40 60 80 1000

20

40

60

80

100

α

W [J

m−2

] Wn Wt Wtot

19

CZ-ELEMENT

Without going into detail about the implementation of the cz-model

in FEM, it has to be noted that special finite elements are devel-

oped and implemented in the commercial finite element package

MSC.Marc/Mentat. Two-dimensional elements for plane strain and

axisymmetry have four nodes and two integration points. A three-

dimensional element with 8 nodes and four integration points is also

available.

20

CZ-ELEMENT

• two-dimensional

plain strain

plain stress

axisymmetric

• three-dimensional

21

FEM : WEIGHTED RESIDUALS

In the deformed state of a material body, the equilibrium equations

must be satisfied in each material point. Together with proper

boundary conditions this vector equation can be solved with the

finite element method (FEM) and is therefor transformed into a

weighted residual integral by use of a vectorial weighting function

~w(~x). Partial integration of the first term leads to a weak ver-

sion with the internal load integral fi as the left-hand side and the

external load integral fe as the right-hand side.

To derive governing FE equations a transformation of the inte-

gral to the initial undeformed state may be helpful. In this transfor-

mation the deformation tensor F is used. The transformed integral

contains the first Piola-Kirchhoff stress tensor T .

The weighted residual integral can be written as the sum of

an integral over bulk material (volume V0b) and an integral over

cohesive zones (volume V0cz).

22

FEM : WEIGHTED RESIDUALS

~∇ ·σ + ρ~q = ~0 ∀ ~x ∈ V∫

V

~w ·

(

~∇ · σ + ρ~q)

dV = 0 ∀ ~w

∫

V

(~∇~w)c : σdV = fe(~w,~t,~q)

~∇ = F−c

·~∇0 → (~∇~w)c = (~∇0~w)c

· F−1

dV = det(F)dV0 = J dV0

∫

V0

(~∇0~w)c· F

−1 : σJ dV0 = fe0

∫

V0

(~∇0~w)c : (F−1· σJ) dV0 = fe0

∫

V0

(~∇0~w)c : T dV0 = fe0

∫

V0b

(~∇0~w)c : T dV0 +

∫

V0cz

(~∇0~w)c : T dV0 = fe0

∑

eb

∫

Ve0b

(~∇0~w)c : T dV0 +∑

ecz

∫

Ve0cz

(~∇0~w)c : T dV0 = fe0

∑

eb

fi0b+

∑

ecz

fi0cz = fe0

23

FEM : TWO-DIMENSIONAL CZ ELEMENT

A cohesive zone element has uniform initial thickness d0. The

length l0 of the element is the undeformed length of the line AB

between the modpoints of the element edges 1-4 and 2-3. Along

this line a local coordinate η is introduced, which spans the range

[−1, 1]. Perpendicular to AB, the local coordinate ξ is defined also

in a range [−1, 1].

In the deformed state a traction ~T works between associated

(= with the same η-coordinate) points P and Q on edges 1-2 and

4-3 respectively. This traction is a function of the elongation of the

line PQ w.r.t. the undeformed state, given by the cohesive zone

law.

24

FEM : TWO-DIMENSIONAL CZ ELEMENT

1

-1

B4

3

12

A

Q

P

ξ~T

η0

fi0cz =

∫

V0cz

(~∇0~w)c : T dV0 = d0

∫

A0cz

(~∇0~w)c : T dA0

~w = δ~u → (~∇0~w)c : T = (~∇0δ~u)c : T = δεl : T

δεl = δεl~e0~e0 = δ~εl~e0

T = T~e0~e0 = ~T~e0

}→ δεl : T = δ~εl ·

~T

=d0

∫

A0cz

δ~εl ·~T dA0 = d0

∫ 1

ξ=−1

∫ 1

η=−1

δ~εl ·~T

h0

2

l0

2dηdξ

δ~εl =δ∆~u

h0

=∆~w

h0

=d0

2

∫ 1

ξ=−1

∫ 1

η=−1

∆~w ·~T

l0

2dηdξ = d0

∫ 1

η=−1

∆~w ·~T

l0

2dη

25

LOCAL VECTOR BASE

The weighting function and the traction are written in components

w.r.t. a local orthonormal basis {~et,~en}, which is defined in the mid-

point of AB. The weighting function components are interpolated

between their values in point A and point B. The interpolation

functions are linear in the local coordinate. The values ∆w in

points A and B are written as the differences of the weighting func-

tion in the nodal points 4 and 1 and 3 and 2 respectively. The index

l indicates that components are taken w.r.t. the local vector basis

{~et,~en}.

26

LOCAL VECTOR BASE

4

3

1

~T

2

A

B

~et

~T ~en

∆~w=∆wt~et + ∆wn~en = [∆wt ∆wn]

[

~et

~en

]

= ∆w˜

T~e˜

∆w˜

T(η)=[

∆wAt ∆wA

n ∆wBt ∆wB

n

]

12(1 − η) 0

0 12(1 − η)

12(1 + η) 0

0 12(1 + η)

=∆w˜

ABTNT(η)

27

LOCAL → GLOBAL

The column with weighting functions is transformed from the local

to the global coordinate system, which is done with a rotation ma-

trix R. It transforms the local base vectors {~et,~en} to the global

base vectors {~ex,~ey}. The only non-zero components in the rotation

matrix are cosine and sine functions of the angle between the line

AB and the global x-axis. The cosine and sine values can be easily

calculated from the coordinates of points A and B : c = (xB−xA)/`

and s = (yB − yA)/`. Coordinates of A and B can be expressed

in the nodal point coordinates : x|yA|B = 12(x|y1|2 + x|y4|3).

28

LOCAL → GLOBAL

∆wAt = w4

t − w1t ; ∆wA

n = w4n − w1

n

∆wBt = w3

t − w2t ; ∆wB

n = w3n − w2

n

∆w˜

AB =

∆wAt

∆wAn

∆wBt

∆wBn

=

−1 0 0 0 0 0 1 0

0 −1 0 0 0 0 0 1

0 0 −1 0 1 0 0 0

0 0 0 −1 0 1 0 0

w1t

w1n

w2t

w2n

w3t

w3n

w4t

w4n

= P w˜

l

w˜

l =

w1t

w1n

w2t

w2n

w3t

w3n

w4t

w4n

=

c s 0 0 0 0 0 0

−s c 0 0 0 0 0 0

0 0 c s 0 0 0 0

0 0 −s c 0 0 0 0

0 0 0 0 c s 0 0

0 0 0 0 −s c 0 0

0 0 0 0 0 0 c s

0 0 0 0 0 0 −s c

w11

w12

w21

w22

w31

w32

w41

w42

= R w˜

29

ITERATIVE PROCEDURE

The internal load integral is the product of column w˜

T and the

column with the internal nodal forces for the cohesive zone element.

The internal load column f˜i is a nonlinear function of the trac-

tion T˜, the element rotation R and the element length `. The

unknown opening of the cohesive zone must be solved from nonlin-

ear (global) equations in an iterative procedure, which means that

each unknown quantity is written as its approximated value ( )∗

and an iterative change δ( ). This is substituted in the internal

load column, which is linearized subsequently. It is assumed here

that δR ≈ 0. The iterative changes of the traction can be expressed

in the iterative opening.

The relation between the iterative traction and the iterative opening

represents the stiffness of the cohesive zone.

The iterative opening is now interpolated analoguously to the in-

terpolation of the weighting function. Also the rotation of the local

to the global coordinate system is applied. The iterative opening

is expressed in the iterative nodal displacements in the global di-

rections. The iterative tractions are then expressed in the iterative

nodal displacements.

The iterative internal load column can now be expressed in the

iterative opening, using the stiffness matrix K∗ of the cohesive zone

element.

30

ITERATIVE PROCEDURE

fi0cz = w˜

T d0 RTPT

∫ 1

η=−1

NT(η)T˜(η)

`0

2dη = w

˜

Tf˜i0cz

(R, T˜)

f˜i0cz

= f˜

∗i0cz

+d0

2R∗T

PT

∫ 1

η=−1

NT(η)δT˜(η) `0 dη

δT˜(η) =

∂T˜

∂∆˜

δ∆˜

=

∂Tt

∂∆t

∂Tt

∂∆n

∂Tn

∂∆t

∂Tn

∂∆n

[

δ∆t

δ∆n

]

= M(η) δ∆˜(η)

δ∆˜(η) = N(η)δ∆

˜

AB = N(η)P δu˜

l = N(η)P R δu˜

f˜i0cz

= f˜

∗i0cz

+

[

d0

2R∗T

PT

∫ 1

η=−1

NT(η)M(η) N(η) P R∗ `0 dη

]

δu˜

= f˜

∗i0cz

+ K∗ δu˜

31

FEM : IMPLEMENTATION

The cohsive zone element is implemented as a ”user element” in

the commercial FEA package MSC.Marc.

At the start of the first increment data are initialized and read

from external files. In the global element loop, the user element

subroutine is called when the element is a cohesive zone element.

At the end of the increment, after convergence, cohesive zone

parameters can be adapted based on the evolution of damage vari-

ables or bulk properties.

32

FEM : IMPLEMENTATION

MSC.Marc/Mentat

FOR EACH INCREMENT

ubginc : inc 0 : init. / decl. comm.bl.

: read user elemnt data

FOR EACH ITERATION

FOR EACH ELEMENT (2x)

ueloop : count elements and cohesive zones

uselem : user element routine

czbehav : element stiffness / internal load

END ELEMENT

END ITERATION

elevar : write integration point output

plotv : write to post file

uedinc : damage evolution

END INCREMENT

33

LARGE DEFORMATIONS

For the two-dimensional cohesive zone elements, the normal and

tangential opening displacements are typically determined by de-

composing the total opening displacement with respect to a local

orthonormal basis {~et,~en} defined with respect to a reference line

in the cohesive zone element. In most cases this reference line is

chosen to be the cohesive zone mid-line, which is the line AB in the

figure below. The choice of the local basis has a pronounced influ-

ence on the decomposition of the opening displacements, resulting

in different normal and tangential openings for different local bases.

In the case of small opening displacements, these differences

can be ignored, but, in the case of large opening displacements this

is not allowed. To verify this statement, an initially undeformed

(= zero thickness) cohesive zone is subjected to large opening dis-

placements by monotonically opening the cohesive zone to the final

geometry as shown in the figure. The normal opening ∆n, normal-

ized by the initial element length l0, for integration point 1 (ip 1) is

shown as a function of the relative time t/τ for three different local

bases. It is clear that the normal opening cannot be determined un-

ambiguously due to the differences in orientation of the local bases.

Its orientation will influence: (1) the magnitude of both ∆t and ∆n,

(2) the coupling between Tt and Tn and thus the total dissipated

energy and (3) the magnitude of the integrated tractions due to a

non-unique choice for the length of the cohesive zone element.

34

LARGE DEFORMATIONS

35

CZ FOR LARGE DEFORMATION

For large displacements it is no longer physical to discriminate be-

tween normal and tangential openings. A large displacement for-

mulation is therefore proposed to resolve the ambiguity induced by

the choice of a local basis, whereby no distinction will be made

between normal and tangential loadings. Instead of defining two

separate constitutive relations for the normal and tangential direc-

tion, only one constitutive relation between the traction ~T and the

opening displacement ~∆ is used. The cohesive zone law is based on

the normal traction relation of the exponential cohesive zone law

discussed before.

A unit vector with ~e is defined between two associated material

points at the interface. The work-of-separation φ is the dissipated

energy after complete opening. The characteristic opening length

δ is the opening for which T reaches the maximum value Tmax.

Upon unloading, the cohesive zone law shows an irreversible

response. Two types of irreversible behavior are considered: lin-

ear elastic unloading to the origin, i.e. elasticity-based damage (i)

and unloading with the initial stiffness of the cohesive zone, i.e.

plastically damaged (ii). Both cases are shown in the figure below.

36

CZ FOR LARGE DEFORMATIONS

0 1 2 3 4 5 60

10

20

30

40

∆ [µm]

T [M

Pa]

~∆ = ∆~e ; ~T = T(∆)~e ; ~T = ~f(~∆) ; T =φ

δ

(

∆

δ

)

exp

(

−∆

δ

)

φ =

∫∞

∆=0

T(∆) d∆ ; Tmax =φ

δ exp(1)

37

MODE-MIXITY

The opening mode of the cohesive zone is quantified by a mode-

mixity parameter d In a two-dimensional cohesive zone, ~d1 and ~d2

are the components of the normals ~n1 and ~n2 of the two cohesive

zone edges perpendicular to ~∆. In a 3D element, ~d1 and ~d2 are pro-

jected on Sn, which is a plane with normal direction ~∆. Parameter

d has a value between 0 (mode-I) and 2 (mode-II). Intermediate

values of d represent a mixed-mode opening. The traction-opening

relation is extended with this mode-mixity d, where a parameter

α controls the influence of mode-mixity behavior: the interface is

stronger (α > 0) or weaker (α < 0) in mode-II than in mode-I.

To investigate the mode-mixity behavior a single cohesive zone

is opened under an angle β and the total dissipated energy is quan-

tified. The value of the potential was taken to be φ = 100 Jm−2.

The two faces are kept parallel to each other and β is defined as

the angle between one of the faces and the vector ~∆.

The results are shown in the figure below. Obviously, the energy

dissipated in mode-I (β = 90o) is independent of α and equals φ.

In the case of mixed-mode or mode-II (β = 0o) opening of the

cohesive zone, the total dissipated energy depends on α. In the

right figure, the traction is plotted for three values of α in mode-II,

where the total dissipated energy (area under the curve) depends

on the parameter α.

38

MODE-MIXITY

d = ||~d1 − ~d2||

T =φ

δ

(

∆

δ

)

exp

(

−∆

δ

)

exp

(

αd

2

)

0 30 60 9080

90

100

110

120

W [J

m−2

]

β [o]

α = 0.2α = 0.0α = −0.2

0 2 4 6 8 100

10

20

30

40

50

∆ [µm]

T [M

Pa]

α = 0.2α = 0.0α = −0.2

39

POLYMER COATED STEEL

Traditionally, products like aerosols, beverage and food cans, beer

caps and luxury products are made from sheet metal. After the

forming steps, the product is cleaned and lacquered to prevent cor-

rosion, to give its surface a glossy appearance and to assure good

printability. Currently, more and more products are made from poly-

mer coated sheet metal, making subsequent lacquering superfluous.

This implies considerable cost savings and eliminates the emission

of volatile organic compounds. Because the coating is subjected

to the same deformation processes as the metal substrate, delam-

ination may occur, leading to the loss of protective and attractive

properties of the product which is unacceptable. If delamination can

be predicted, the processing routes, parameters and tooling can be

adjusted to prevent it.

40

POLYMER COATED STEEL

41

CAN MAKING

Can making starts with the deep-drawing of a cup. In a number

of subsequent steps the can wall is ironed between two circular

dies of decreasing diameter. The can becomes longer and the wall

thickness is reduced.

The ironing process does hardly damage the polymer coating,

due to the ocurring high pressures. Deep-drawing, however, where

pressures are much smaller, may lead to delamination.

42

CAN MAKING

43

MATERIAL SYSTEM

The material system is schematically shown in the figure below.

The substrate is a batch annealed deep-drawing steel. The steel

is coated with chromium in order to improve the adhesion of the

polymer layer. The polymer coating consists of two layers: a thin

adhesion layer and a significantly thicker poly-ethylene terephtha-

late (PET) layer. The overall thickness is 30 µm. The adhesion

layer is made of PETG (which is a modification of PET with glycol

side groups) and several additives to make it stronger.

Adequate constitutive models for coating, substrate and the in-

terface have to be used and their (material) parameters have to be

determined through dedicated experiments. The steel substrate is

modeled as an elastoplastic material. The yield stress is 250 MPa,

the ultimate (engineering) tensile stress is 310 MPa and the (engi-

neering) linear strain at break is approximately 25 %. These data

are measured in standard tensile tests together with the hardening

curve. The behavior of coating is described with a nonlinear vis-

coelastic material model, which will not be described here. Various

parameters of this model could be determined in separate experi-

ments, performed on thin polymer sheet material. However, some

of the parameters could only be determined together with the co-

hesive zone parameters, during peel tests.

44

MATERIAL SYSTEM

”material” model parameters

steel elastoplastic E = 210 GPa ; ν = 0.3

σy0 = 250 MPa

experimental hardening curve

coating viscoelastic various

interface cohesive zone δ ; φ ; α

45

SPECIMENS AND EXPERIMENTS

Specimens of 40 × 8 mm2 were cut out of the polymer coated

sheet. Secondly, the width of the specimen was reduced to 3.5 mm

by grinding the edges with SiC-paper, which also removed possible

burrs. Next a groove was milled in the back of the specimen with a

table top precision cut-off machine. The specimens were clamped

in a standard tensile tester and loaded uniaxially with a speed of

30 µms−1 until the steel substrate fractured at the location of the

groove.

The test set-up for the zero degree (0o) peel tests and 90

degree (90o) peel tests is shown in the figure below. The 90o peel

specimens consist of two peel specimens of which half the lengths

are adhesively bonded. The bonded part is placed vertically between

to parallel steel plates, allowing vertical displacements only. The

advantage of this set-up is that the peel angle remains consistently

equal to 90 degrees. The displacement is prescribed with a constant

velocity of ux = 25 µms−1.

46


47

EXPERIMENTAL RESULTS

Both peel tests were done with a micro tensile stage, mounted in

an ESEM, which allowed observation of the delamination. In the

90o peel test, fibrillation was observed as shown in the figure below.

This setup also allowed the measurement of the characteristic length

δ.

During the experiments the force and displacement are moni-

tored. The normalized peel force as a function of the clamp dis-

placement is shown in the figure below for both the 0o and 90o

peel tests. The average normalized peel force for the 0o peel test

is F∗0 = 4.85 N with a standard deviation of 0.07 N. For the 90o

peel test the average normalized peel force is F∗90 = 4.16 N, with

a standard deviation of 0.06 N. The scatter in the experimental

results is caused by small variations in experimental conditions and

possible variations in specimen width, coating thickness, ambient

temperature and relative humidity. Considering all these sources of

scatter, the results seem remarkable consistent.

48


49

PARAMETER IDENTIFICATION

Parameter values were determined in a combined experimental-

numerical procedure, where the tests were simulated and the nu-

merical results were fitted on the experimental data. Two models

(3D and plane-strain) are created in a finite element solution con-

text. Due to symmetry conditions, only half of the experimental

set-up needs to be modeled. The 3D model is discretized with

eight-node hexahedral elements and the plane-strain model with

four-node quadrilateral elements. The cohesive zones are located

between the coating and the substrate and have an initial zero

thickness.

A mesh size dependency study has been performed and it was

concluded that the maximum element size (in peel direction) is 1

µm and the maximum load step size is 100 nm per increment. The

determination of the interface or material parameters through the

adopted parameter identification procedure requires a sequence of

simulations to be carried out. Using 3D peel simulations to this

purpose is computational expensive. Therefore, it was investigated

whether a peel test model with a plane-strain geometry could be

used instead. Several 3D simulations, with different widths, have

been carried out and results have been compared to plane strain

simulations. It could be concluded that the difference in results was

acceptable, since it falls well within the experimental measurement

errors. Parameter identification simulations are therefore carried out

with a plane strain geometry to significantly reduce the calculation

time, typically with a factor 15-50.

50


φ = 194 Jm−2 ; δ = 1µm ; α = 0

51

INFLUENCE OF SUBSTRATE ROUGHNESS

To investigate the influence of roughening on the adhesion, speci-

mens were predeformed to 5%, 10%, 15% or 20% strain by uniaxial

stretching with a clamp speed of 50 µm/s. The loss of adhesion

is characterized by a damage variable ω, which is taken to be a

function of the effective plastic pre-strain εp. The influence of

roughening enters the traction-opening law through the potential

φ. The potential value for an undamaged interface is indicated as

φ0. A parameter identification procedure resulted in a quadratic

function for the relation ω(εp). The maximum used (uniaxial) pre-

strain is 0.2, because higher strains result in strain localization and

subsequent failure of the specimen. In an industrial deep-drawing

process the maximum effective plastic strains are typically in the

order of 0.5-0.7, where the effective plastic strain is composed of

multi-axial strains that are induced by non-homogeneous loading

paths under complex boundary conditions.

52

INFLUENCE OF SUBSTRATE ROUGHNESS

undeformed 20% pre-strain

φ(εp) = [1 − ω(εp)]φ0

ω(εp) = γ1 ε2p + γ2 εp with γ1 = 0.913 and γ2 = 0.147

53

DEEP-DRAWING

The cohesive zone model with the fitted parameter values, was used

in an axisymmetric model to simulate the deep-drawing of a circular

blank. The initial geometry and the element discretization is shown

in the figure below.

After some loading steps, the deformation is shown in the figure

below. Delamination was found to occur in the incoming bend of

the sheet (location d).

54

DEEP-DRAWING

55

PARAMETER VARIATION

The influence of several deep-drawing parameters on the interfacial

integrity is investigated. The results of the these simulations are

assessed by an interfacial integrity parameter κ, which is defined

as the fraction of energy that still can be dissipated at he interface

before complete debonding takes place. The maximum achieved

opening displacement of a cohesive zone during its loading history

is ∆m. The initial value of κ is obviously 1, whereas complete

interfacial failure corresponds to κ = 0. The minimum value en-

countered in the model is plotted against varied parameters in the

figures below.

56

PARAMETER VARIATION

κ =Wrest

φ0

with Wrest =

∞∫

∆m

T(∆) d∆

57

EASY PEEL-OFF LID

Many food containers are sealed with a so-called Easy Peel-Off Lid

(EPOL). The EPOL is easy to open and to produce and consists

of two parts: the protact ring and the alufix. The protact ring

is made of polymer coated steel with a PolyPropylene (PP) layer

at the outside and a PolyEthyleneTerephthalate (PET) layer at the

inside of the can. The alufix is a membrane, consisting of aluminum

which is coated with a peelable PP layer at one side of the foil. The

thermoplastic PP layer of the protact ring and alufix foil are heat

sealed together, achieving an airtight closure.

The EPOL closure must be peelable with a force not exceeding

25 N under an angle of 135o, which is the industrial standard for

opening peelable food containers. The EPOL closure must also be

airtight and resist a pressure build-up in the can of about 2.4-2.6 bar

at sterilization temperature, which also is a common requirement

in the industry.

58

EASY PEEL-OFF LID

59

MATERIAL SYSTEM

The protact part is a polymer coated steel, which has a PET layer at

one side and a PP layer at the other. The PET layer has a corrosion

protective function and the PP layer is used for heat sealing and

also protects the steel against corrosion.

Three protacts are investigated:

• P1073: with a substrate thickness of 0.21 mm and a PP layer

of 25 µm.

• P1074: with a substrate thickness of 0.21 mm and a PP layer

of 40 µm.

• TP823: with a substrate thickness of 0.18 mm and a PP layer

of 40 µm.

The distinctness of the TP823 with respect to the other two is that

the PP layer contains TiO particles to give the PP a white color for

aesthetic reasons.

Aluminum foil for non-aggressive products is used in this study.

It has a substrate thickness of 60 m and a peelable PP layer of

approximately 20 µm.

The steel and alufix are modeled as elastoplastic materials, using

measured material parameters, which are listed in the table. For

alufix, Young s modulus, initial yield stress, ultimate stress and

Poisson’s ratio are determined for rolling direction (0o) and for the

direction perpendicular to this (90o). Hardening curves were also

recorded.

The PP layer is not modeled, so the cohesive zone reflects the

interaction between alufix and substrate.

60

MATERIAL SYSTEM

Material E [GPa] σy [MPa] σUTS [MPa] ν [-]

Steel 210 340 370 0.3

alufix 0o 37 53.6 117 0.33

90o 30 52.5 103 0.33

61


A numerical-experimental approach is followed to determine the

cohesive zone parameters, where peel-off experiments were done

and simulated, after which parameters were determined using a fit-

ting procedure. For this purpose strip specimens were made and

tested in 90o peel-off experiments. The peel-off speed for all mea-

surements is 160 µm/s, which is the maximum speed of the used

tensile machine.

62


63


The measured force-displacement response shows an initial peak in

the peel-off force followed by a plateau region. A polynomial curve

has been fitted onto the experimental data in a least squares sense.

64


65


To investigate the behavior of the EPOL closure, a finite element

model must been made, where an interface crack between alufix and

steel substrate is propagated to simulate the opening. The interface

crack is modeled with a cohesive zone element. The cohesive zone

model does not describe mode-mixity.

The characteristic length δ is measured from ESEM images,

recorded during the peel-off experiment. Although the accuracy

of the values can be doubted, it appeared that the sensitivity of

the value is not large. The potential φ is fitted onto the plateau

force value, which was measured and also simulated with the finite

element method using a model of the peel-off test.

A mesh sensitivity analysis is performed for the validation model.

The number of elements over the thickness of the aluminum foil was

chosen to be five, which appeared satisfactory. The number of el-

ement over the length of the strip is important because this also

includes the cohesive zone elements which are responsible for the

peel-off force. It appeared that using 200 elements was accurate

enough.

The parameter values are listed in the table for the three pro-

tact materials. Also the measured and simulated force-displacement

curves for P1073 are shown.

The peel-off specimens are also tested with a 135o peel-off load.

Using the fitted data from the 90o peel-off tests, these experiments

are also simulated and both results are shown in the figure for

P1073.

66


Protact φ [jm−2 Tmax [MPa] δ [µm]

P1073 170 19.5 3.2

P1074 213 11.7 6.7

TP823 162 19.9 3.0

P1073 : 90o P1073 : 135o

67

BURST PRESSURE EXPERIMENTS AND SIMULATION

The food sterilization process, where food is heated up for a specific

time at a temperature of 121o C, takes place inside a closed can.

During this process the pressure inside the can increases, which may

lead to EPOL burst. At Corus burst pressure experiments were done

at room temperature (20o C) and at sterilization temperature (121o

C). For all material systems, the alufix foil ruptured before failure of

the adhesion interface. At room temperature the maximum pressure

was 3.4 bar and at elevated temperature the maximum pressure

was 2.3 bar on average. This premature alufix failure prevented

the validation of the interface crack propagation model by burst

pressure simulations. To make some comparison, the deformation

of the protact rings were measured and used later for comparison

with numerical results.

Using the fitted cohesive zone parameters and other material

and geometric data, the burst pressure experiments were simulated.

Obviously no failure of the alufix could be implemented, so the

calculated burst maximum pressures at seal failure are far too high:

typically 3.1-4.3 bar. In the model, the residual stresses, strains

and hardening from the forming of the protact ring are taken into

account. The deformation of the protact ring could be compared

to the measured deformation and is shown in the figure for TP823.

Also a detail of the deformed cohesive layer is shown at a pressure

of 5.0 bar.

68

BURST PRESSURE EXPERIMENTS AND SIMULATION

69

OPENING EXPERIMENTS

The EPOL peel-off experiments are performed at Corus, where a

special EPOL peel-off tester is available. For P1073 protact, the

force-displacement curve is shown in the figure. The higher loads at

the start and the end of the experiment is caused by the geometry

of the seal.

70

OPENING EXPERIMENTS

71

OPENING SIMULATIONS

The EPOL peel-off model, is a three-dimensional FE model, which

uses the cohesive zones to describe the interfacial fibrillation of the

polymer layer during peeling. The model is reduced to one fourth

of the EPOL to diminish computing effort. This model reduction is

justified because the first half of the peel-off response of an EPOL

contains information about the initial peak force and the force in

the mid section, which is of most interest.

The deformation of P1073 protact is shown in the figures at

various stages of opening. Also the force-displacement curve is

depicted and compared to the experimental results.

72

OPENING SIMULATIONS

73

PARAMETER VARIATION (P1073)

Some parameters were given different values and the result of these

variations on the opening force-displacement are shown in the fig-

ures. In the first figure the width of the peal-off lid is varied. In

the second figure four simulations with different values for the seal

angle are presented. In the third figure three different seal widths

are used.

All simulatioons were done for the P1073 Protact material.

74

PARAMETER VARIATION (P1073)

lip width / seal angle / seal width

75

SOLDER JOINT FATIGUE

Solder joints provide electrical, thermal and mechanical continuity

in electronic packages. Today, miniaturization is the major driving

force in consumer electronics design and production. Efforts in

decreasing component dimensions have led to the development of

ball grid array (BGA) and flip chip packages, where solder balls are

employed. Following the legistative ban on lead (Pb), new soldr

alloys have been developed recently, which consist mainly of tin

(Sn).

76

SOLDER JOINT FATIGUE

SAC

top-Cu

bot-Cu

silicon

FR4

Y

Z X

1

-7.155e-05

4.536e-04

9.787e-04

1.504e-03

2.029e-03

2.554e-03

3.079e-03

3.604e-03

4.129e-03

4.654e-03

5.180e-03

lcase1

Total Equivalent Plastic Strain

Inc: 25

Time: 5.000e+00

Y

Z X

1

SnPb → SnAgCu

77

(THERMO)MECHANICAL CYCLING → FATIGUE LIFE

These solder balls are subjected to different types of loading:

• Thermal cycling due to repeated power switching evokes heat

related phenomena: the mismatch in the coefficient of thermal

expansion between the package components causes cyclic me-

chanical strains.

• As a result of the multi-phase nature of Sn based solder alloys

and the thermal anisotropy of β-Sn, internal stresses build up

in the solder.

• Cyclic thermo-mechanical loading evokes creep-fatigue damage

or creep rupture.

• Bending of the board induces shear and tensile stresses in the

solder joints.

The figure below shows a typical bump between two pads, where

thermal loading of the system leads to deformation and results in

plastic strains. Cyclic loading leads to fatigue damage and finally to

failure. Instead of using phenomenological laws like Manson-Coffin,

the fatigue damage is assessed with a cohesive zone approach.

78

(THERMO)MECHANICAL CYCLING → FATIGUE LIFE

SAC

top-Cu

bot-Cu

silicon

FR4

Y

Z X

1

-7.155e-05

4.536e-04

9.787e-04

1.504e-03

2.029e-03

2.554e-03

3.079e-03

3.604e-03

4.129e-03

4.654e-03

5.180e-03

lcase1

Total Equivalent Plastic Strain

Inc: 25

Time: 5.000e+00

Y

Z X

1

• Manson-Coffin (phenomenological, SN curve fitting)

∆εp = ε ′f (2Nf)

c → Nf = 12

(

∆εp

ε ′f

)1/c

– no micro-structure

– no damage initiation and propagation

⇒ no information for redesign bump shape or joint layout

better fatigue damage prediction needed

79

COHESIVE ZONE FOR FATIGUE FAILURE

The interaction between two associated points of a delaminating

interface in the solder joint is given by a traction-opening law :

Tα = Tα(∆α), where ∆α is the opening and Tα the traction, either

in normal (α = n) or tangential (α = t) direction. Because

deformations are small, there will be no problem relating to the

decomposition in normal and tangential direction. The relation

between traction and opening is taken to be linear. Damage is

introduced in the model by the degradation of the stiffness from a

given initial value to zero. The damage is quantified by a damage

variable Dα, which evolution during the loading history is given by

a damage evolution law. Its initial value is theoretically zero and its

final value at complete failure is one. The initial stiffness kα has

to be taken large enough to prevent any influence of the cohesive

zone layer, when there is no damage. Obviously, this initial value is

reversely related to the thickness of the cohesive zone, approaching

infinity, when this thickness is zero. The damage evolution law will

prevent damage growth, when the absolute value of the traction is

below the fatigue limit σf.

80

COHESIVE ZONE FOR FATIGUE FAILURE

Tα = kα(1 − Dα)∆α

Dα = cα|∆α| (1 − Dα + r)m

⟨

|Tα|

1 − Dα

− σf

⟩

kα MPa/mm initial stiffness

cα mm/N damage growth coeff.

r - coefficient

m - exponent

σf MPa fatigue limit

−5 0 5 10 15 20x 10−5

−400

−300

−200

−100

0

100

200

300

400

∆n [mm]

T n [M

Pa]

−2 −1 0 1 2x 10−4

−150

−100

−50

0

50

100

150

∆t [mm]

T t [M

Pa]

81

CHOICES FOR DAMAGE GENERATION

The damage evolution is described by the evolution equation. Addi-

tionally it can be decided, whether damage grows during the whole

cycle or only during a confined stage. The figure shows the influence

of this choices for the damage growth in one cycle.

82

CHOICES FOR DAMAGE GENERATION

• {∆t→0, ∆n→0, ∆n<0} = { 0|1, 0|1, 0|1 }

0 2 4 6 8 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2 x 10−4

time

u y

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

timeD n

• penalty for compression

• k(1 − D) = kf → D = 0

83

MATERIAL SYSTEM

A typical solder joint is shown in the figure below. The solder is an

alloy of Sn (about 95%), Ag and Cu, referred to as SAC. The pads

on both the chip side and the board side are made of thin metallic

layers of Cu, Ni and Au, the so-called under-bump metallization

(UBM). In the interface region, inter-metallics are formed, which

have a big influence on the (fatigue) behavior.

Material parameters of the bulk materials – pads, solder, board,

silicon – were determined with large scale tests and small scale

indentation experiments and are listed in the table below.

The solder ball is not homogeneous as can be seen in the figure

below. It consists of a few grains, which have different orientations

of their lattice. Because Sn is anisotropic, this inhomogeneity will

lead to strains, stresses and fatigue failure, when a cyclic thermal

loading is applied.

The fatigue damage is predicted by the cohesive zone law,

which is applied at the SAC/pad interfaces and at the grain bound-

aries. Parameters will be different for the two interfaces.

84

MATERIAL SYSTEM

material E ν α

[GPa] [-] [ppm/oC]

SnAgCu 64.1 0.4 16.5/32.4

Cu 128 0.35 17

Ni 197 0.31 12.96

Ni3Sn4 140.4 0.3 14.98

Cu6Sn5 94.1 0.35 17

Ag3Sn 55.3 0.35 16

board (FR-4) 17.5 0.11 17.6

Molding Cmpd. 10 0.25 16.9

substrate 1 0.11 16

Solder Mask 2.412 0.467 60

85


To test the fatigue behavior of the interface, tensile and shear spec-

imens were made. Both tensile and shear experiments/simulations

were done, using a micro tensile stage, mounted in an ESEM, to

allow for observation of the interface.

To investigate the degradation of the inter-granular interface

in the bulk solder, unconstrained test specimens were subjected to

a cyclic thermal load. The resulting loss of interface integrity was

assessed by measuring the Young’s modulus after a certain number

of loading cycles.

86


mechanical cycling → reaction force/area (nr. cycles)

SEM, OIM, misorient.

500 cycles

−40 < T < 125 oC

thermal cycling → macroscopic stiffness

87

PARAMETER IDENTIFICATION : SAC/PAD : MECHANICAL

Assessment of the parameters in the damage evolution law has

been done with an experimental-numerical approach, where fatigue

failure data from experiments were confronted with simulation data.

The tensile and shear experiments have been modeled with the finite

element method and these models have been subjected to a cyclic

mechanical load.

The parameter identification procedure resulted in values for

the cz-parameters, which are listed in the table below. Experimental

and numerical results are also shown in the figure as stress against

the number of loading cycles.

Also the bulk solder has been modeled and subjected to the

cyclic thermal load from the experiments. The cz-parameters for the

inter-granular interfaces were fitted such that the stiffness reduction

of the model was the same as that of the experiments.

88

PARAMETER IDENTIFICATION : SAC/PAD : MECHANICAL

cz k c m r

[N/m3] [m/N] [-] [-]

SAC/pad n 8.79e8 68000 3.160 1e-6

SAC/pad t 3.21e8 47000 3.135 4e-5

89

PARAMETER IDENTIFICATION : SAC GR.B. : THERMAL

The parameters of the intergranular cohesive zones can not be de-

termined by mechanical testing. Instead a thin sheet of solder ma-

terial is loaded with a cyclic temperature. Due to the anisotropic

thermal and mechanical properties of the various grains, stresses in

the material occur, which constitute load on the interfaces. This

cyclic load results in damage growth. The damage as a function of

the number of cycles is characterised by the global Young’s modu-

lus of the material. Confronting experimental data with numerical

results allows the parameter identification.

90

PARAMETER IDENTIFICATION : SAC GR.B. : THERMAL

cz k c m r

[N/m3] [m/N] [-] [-]

SAC grain.bndy. n 6.40e8 42000 2.940 0

SAC grain.bndy. t 2.28e8 42000 2.940 0

91

SOLDER BALL FATIGUE FAILURE

Having determined all parameters of bulk materials and cohesive

zones, the damage evolution in a bump could be analyzed. The fig-

ure below shows a two-dimensional plane strain bump/pad model.

Boundary conditions are very important for the fatigue life, so dif-

ferent load cases have been studied.

The damage in the solder ball and the solder/pad interface is

shown in the next figures after 250 and 1000 cycles, for the three

selected boundary conditions. The deformation is enlarged 5×. It

is clearly seen that the bump is completely detached from the pad

after 1000 cycles.

92


PSfrag replacements

cz1cz2

cz3cz4

PSfrag replacements

X

Y

Z

1

X

Y

Z

1

X

Y

Z

1

X

Y

Z

1

X

Y

Z

1

X

Y

Z

1

93


The simulation results are validated using an experimental analysis

of BGA packages under thermal cycling and thermal shock loading.

All slice models are computed separately for both types of loading,

until N = 1000 cycles. The numerical results are then extrapolated

to N = 5000. A critical effective damage value Deff is defined

to predict the number of cycles to failure (ncf), that adequately

reproduces the experimental failure distributions provided by the

industry.

First, all slice models were computed without any defects, rep-

resenting a theoretical case. Then, defects were introduced in the

mesh, as they were statistically determined. Last, a data set con-

sisting of good (45 %) and defective balls (55 %) is constructed.

The figure shows that, with a critical damage level for failure

equal to Dcriteff = 0.87, a good agreement between the statistically

constructed data set and the experimental data is obtained. The

fatigue life of long living (N > 5000) solder balls is overestimated.

This seems logical due to the fact that, as mentioned previously,

the cross-sectional defect analysis hides many defects, yielding too

optimistic results (45 % of all solder balls were defect-free). In

reality, more defects have to be inserted into the slice models.

94


cycling shock

95

PARAMETER VARIATION

Although many parameters could obviously be varied, the behavior

of various solder geometries and grain distributions was investigated

when they were subjected to cyclic mechanical loading. The solder

joint is placed between chip and board as is shown in the figure.

From these simulations the following conclusions could be drawn:

• As the contact angle α decreases, cross-sectional averages of

plastic strain and Von Mises stress values increase.

• A horizontal grain boundary becomes more critical when the

contact angle increases. However, the highest stress is found at

bump/pad interface.

• A vertical grain boundary may play a crucial role in solder joint

reliability, which becomes more pronounced as the grain bound-

ary area increases.

• Vertical grain boundaries that touch the bump/pad interface

have the highest stress level under thermal loading and there-

fore constitute a possible crack initiation path starting from the

bump/pad interface.

96

PARAMETER VARIATION

• α ↓ → (εp, σvm)crossection ↑• α ↑ → hor. grain boundary more critical

• gr.bnd.area ↑ → vert. grain boundary more critical

• vert. grain bnd’s touching bump/pad →highest stress level

97

END

98

END

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