Procesos cognitivos en la lectoescritura

4.7.1 Mechanics of Interfacial Delamination

A fracture may start in an interface but will only propagate along it if that is the path of least resistance. The most favourable path depends upon both the energy release rate and the interfacial toughness, which is strongly influenced by mode-mixity. However, in this section we will only consider fractures that do not kink out of the interface.

For any bimaterial system, in which the materials are considered as isotropic and linear elastic, four elastic constants, i.e., two Young’s moduli and two Poisson’s ratios are involved. However, it has been proven that under traction-specified boundary conditions the solution to plane problems of elasticity depends on only two-dimensional combinations of the elastic moduli, namely, Dundurs parameters defined by [17]

1 1 2

2

1 1 2 2

/ ) 1 ( / ) 1 (

/ ) 1 ( / ) 1 (

µ ν µ

ν2₂))// 2₂ ((11 1₁)/ µ ν µ

ν2₂))// 2₂ ((11 1₁)/ α ₂) /)) /)// ₂ (((111

1 ( / ) /

) (1

= (116)

µ ν µ

ν

µ ν µ

β ν

/ ) 2 1 ( / ) 2 1

( ν µ νν

/ ) 2 1 ( / ) 2 1

( νν µµ νν

2 1

νν1 2

2 µµµ ν µ ν

1

1 µ

ννν

2 2 µµµ ν µ ν

1 ( / ) 2 )/ 2νννν µµµµ 2ν µµ

= (117)

or by the connection,

β β ε π

+

= − 1 ln1 2π

1 (118)

where νν is Poisson’s ratio,µµ is shear modulus, and subscripts 1 and 2 refer to materials across the interface. The αα measures the relative stiffness of the two materials. Material 1 is stiffer than material 2 if α > 0. The so called oscillating index, ββ ^or εε, which is responsible for oscillating stress behaviours at crack tip in some cases as to be discussed shortly, can hardly be interpreted intuitively. In general, thoughβ≠0 it is usually small (< 0.25), and often approximated to zero. Whenβ = 0, the singularity at the tip of an interface crack has the same as that in an isotropic homogeneous solid, and the normal definition of stress intensity factor can be used. The only difference being that stress distribution on an interface is usually mixed mode. If β ≠ 0 the stress ahead of a crack tip oscillates and there can be contact between the interface surfaces. As Rice [18] pointed out, the most

difficult problem that the oscillatory stress distribution presents is in understanding the mode-mixity which varies with distance ahead of the crack tip. The stress intensity whenβ ≠ 0 is most usefully defined following Rice, who defines the stresses, a distancerrahead of the crack tip as

r i Kr

i

σ π

σ ^ε

2π

12

22 σ

σσ +ⁱiⁱⁱσσ12 = ⁽¹¹⁹⁾

where K is the K complex stress intensity factor with dimension FL^–3/2L^-iε . The complex stress intensity factor K has real and imaginary components (K ((K1,KKK2) which have a similar role to the mode I and II stress intensity factors in homogenous materials to which they degenerate when β = 0. The energy release rate in terms of the stress intensity factor components is

2 2

2

2 2

2

1 2

1 2

1 ²

1

1 (( ₁² ₂²)

2

( )

( ₁

G 1

(

^{1 1 2}

)

E₁ E₂ ν

2 1

1 1

1

β² 1 ¹

− 2

( 1₁

(

^{1 2}

)

⁽¹²⁰⁾

Many experiments have shown that the interfacial toughness depends strongly on the mode mixity. Figure 18 shows the effect of mode-mixity on a microelectronic interface. Near the tip of an interfacial crack a local, position dependent mode-mixity is defined by σ12/σ22 and what has been termed the local phase angle,ψ^local, is given by

) / ) ln(

Re(

) tan Im(

tan ¹

22 1 12

local _i //

i

ε σ ψ

ψ σ _ε^ε »=ψ +

¼»»

º»»

«¬

««ª

= ««

»¼

»»º

»»

«¬

««ª

= ⁻ «« ⁻ (121)

where L is material characteristic length or a specimen length such as film thickness. The former definition is more appropriate to the effect of mode-mixity on interface toughness. Since ε is small, for over a hundred material pairs studied by Suga et al. [19], the variation in mode-mixity angle with the chosen characteristic length is not large. The characteristic length must be greater than the contact zone for equation (119) to be valid. Rice has shown that if the radius of contact is to be less than 0.01Lthen the phase angle must be greater than –69 degree if ε= 0.08. This restriction will usually be met in practice. There is little accuracy lost by assuming β= 0.

Figure 18. Toughness dependency on the phase angle

4.7.2 Stress Singularity of a General Bimaterial System

In spite of complexity of geometry and material combinations, all stress singularities arising in bimaterial wedge configurations in electronic packages can be grouped into two categories, i.e., an angular corner of bimaterial wedge, and a bimaterial wedge with adhesion, as shown in Figure 19 (a) and (b), respectively [16].

θI

θII

(a)

θW

(b)

Figure 19. General cases of bi-material wedges

The singular stress field is generally expressed as following

1 )

rp^{p 1}f ( ,p r^{p 1}f ( ,

ij ffij

σ θ ⁽¹²²⁾

where r, and θθ are polar coordinates at wedge tip, and p the order of singularity. If 0 <p< 1, the stress field is singular. The order of singularity p

is not only dependent on the material properties (α and β or ε), but also dependent on the geometry, such asθθθ^{I and}θθθ^{II, or} θ^W for two different cases in Figure 19. It should be noted that equation (122) is valid only when p is a real value. In some cases,p appears to be complex value, which introduces oscillating stresses in the vicinity of wedge tip. The stress field in this case should be written in the form of σ_ij =r^RRep11fff_ij(( ,,p,,ImImplnlnr). The term Implnr gives rise to the oscillating behaviour of stresses, but the stress singularity still remains as Re(p(( -1). In the following sections, the determination of the singularity order pis outlined for two cases respectively.

Figure 19(a) shows a general configuration for an angular corner of a bimaterial. The configuration is reduced to a standard interface crack, which refers to a crack lying along the interface in dissimilar material (θ^I=π, θ^II= π). A free edge of bimaterial corresponds toθ^I =π/2,θ^II =π/2. With θ^I = π/2, θ^II = 3π/2, the configuration corresponds to a corner (e.g., chip/underfill in Figure 1) with delamination along one interface.

The general characteristic equation for determining the order of the singularity is

2 2 2 2

2 2 2

2 2 2 2 2

2 22 22 22

2 2 2 2

2 2 2 2

2 2 2 2

16[sin (² )) ²sinsin ][sin (][sin ( ) sin²² ]

16 ²{sin²² [sin (²²² )) ²²²²sinsin²² ] sin² [sin (²² ) ²²sin² ]}

{16 ²( ²² 1) sin² sin

4[ ²sin (² )) sinsin²² (( )]})]}

16 ²[sin² sin ( )) sinsin

I I

2 2 2

sin ][sin (

2 2 2

sin ][sin (

2 2 2 II

2 2 2 2 2 2 2

2 III

II I 2 III

II I II I

I 2(

2(

2 I 2 I

) i

I 2

) sin

p)))) p p

p

{s [s (

{sin [sin ([s ( )) ((

s ( ) s

sin ( )) sins p [s[sin ssin (s (( p

θ θ θ

θ β

θ θ θ

θ θ θ αβ

θ θ

θ θ θ θ α

θ θ θ

−

+16 ²{sin{sin² [sin ([sin ([sin ([sin (²²²²²² ) +sin² [sin ([sin (²²² )) +{16 ²((( ²

−4[ ²sin (sin (² ))))) sinsinsinsin² (((

+16 ²[sin[sin² sin (sin (sin (sin (²²( )) ²

2 2 2 2 2

2

2 2 2 2 2

2 2 2

sin ( )]

8{ 2 ²[sin² sin (sin (² )) sinsin sin (sin ( )]

[sin (² )

sin² ]] [sin ([sin (²² )) ²²sinsin²² ]}]}

4 ²sin (² )) 4 sin4 sin²² (( ) 0

I I

II 2

sin (²

I 2(

2(

2 I 2

) i ²

I ) sin^{2 222}(( ^I

II

II I I

I II I II

p ) sin sin ( ) sin sin ( ) sin sin ( p

p sin ]] [sin ([sin (

p ssin (( ))) 4 sinss p

θ β

θ θ θ θ

θ

θ θ θ α

θ θ θ θ

+8{

+

−p sin ]] [sin ([sin ( )

−4 ²sin (sin (² ))))) 4 sin4 sin4 sin4 sin4 sin² (((( )

(123)

The solution can be obtained by solving this equation numerically by Newton-Raphson method. In the case of a standard interface crack, i.e., θI= π,θ^II= π, the well-known complex stress singularity is found, see Rice or equation (119) with

p = 1/2+iε (124)

The imaginary part of the stress singularity at above equation introduces oscillating stresses in the vicinity of the crack tip and a possible overlapping of both materials. However, in the case of θθθ^{I +}θθθ^{II< 2}π, stress field around the tip does not exhibit an oscillating singularity. For instance, at the free edge of bimaterial, the stress field is singular, but showing no oscillating behaviour.

Figure 19(b) shows a bimaterial wedge in which two bonded interfaces are involved. The characteristic equation of the order of singularity for this situation is derived by ref. [16]

0 ))}

( ( sin ) 1 )(

1

( )(1 )sin( ( )

)) 2 ( sin(

) sin(

) )(

1

{( )( )sin( )sin( ( 2 )

))) ( ( sin ) 1 ( 2 )) 2 ( ( sin )

( ) sin( ( 2 ) )sin( ( )

) ( sin ) 1 {(

sin ) 1 ( sin ) 1

( {(1 ) sin( )

}2 2 }

n

2 n

1

2 1

n2 2 n

2 2sinsinii

2

2 2sinsinii

1 2

2 ii

2 4 4 ii

4

))}

Γ )

− (1(1 )(1 +{(1 )((

1 ( 2 )) 2 (1 ) +( 2

sin ) 1 ( sin

) sin (1 )

) sinsin sinsin {(1

π θ κ

κ

π θ θ

κ κ

π θ κ

π θ κ

θ κ

θ λ

θ

( )

( ) (

( )sin(

( )sin()

p p

p

(125)

where

¯®

¯¯

­®®

= (3− )/(1+ ) planestress strain

plane 4

3−

1 1

1

1 ν ν

κ ν

¯®

¯¯

­®®

= (3− )/(1+ ) planestress strain

plane 4

3−

2 2

2

2 ν ν

κ ν (126)

1 2/µ µ

= Γ

It was found that the order of stress singularity is always considerably smaller than the same wedge system with one delaminated interface (obtained from equation (124), with θ^I + θ^II = 2π). This implies that the stress is released when delamination is present at one interface for a bimaterial wedge. For most cases in Figure 19(b), the stress field does not have an oscillating singularity.

4.7.3 Delamination Criterion for Interfaces at Bimaterial Wedge For a crack in homogeneous material, the well-known mode I, II, and III stress intensity factors (KI, KII, and KIII) were found to be the only parameters that characterize the magnitude of the singular stress field at the

crack tip for each basic failure mode, respectively. For a standard interface crack in a dissimilar material (i.e., θ^I =π,θ^II= πin Figure 19(a)), the total energy release rate or J-integral can be used as fracture parameters, while the separated mode fracture parameters like KI and KIIdo not exist except for special material combinations with β= 0. For a bimaterial wedge with two bonded interfaces, however, it is impossible to find a unique parameter (or a few) to represent the whole singular field of the wedge tip. In the following analysis, without lack of the generality, a bimaterial wedge with two bonded interfaces, which are placed horizontally and vertically, respectively, is considered (see Figure 20). The singular stress field for this situation can be represented in the form of equation (122). The singularity shows no oscillating behaviour and the order is much less than ½. With the introduction of the polar coordinate system, the hoop stressσθ and τ^rθat the tip along the different angular direction can be expressed as

p

r g r

r

f( )/ ¹¹ ,, _r g( )/ ¹⁻

σθ = ^f r^1−p, θ =g(( ) (127)

Unlike the stress field in a crack tip of a homogeneous material, where the stress intensity factors (KI, KII, and KIII) are found to be only parameters governing the crack tip behaviours, equation (127) virtually gives no such kind of solution. However, the opening and shearing mode ‘stress intensity factors’ can be defined as function of the angular position at the corner tip by

) ((

g K f

K_o (( ), _s (128)

The introduction of opening and shearing stress intensity factors in Equation (128) is important, since in the mixed-mode crack propagation of homogeneous material, the cracking is present in the direction of Ks= 0 or maximum Ko. For a bimaterial wedge considered here in Figure 20(a), the opening and shearing stress intensity factors along the two bonded interfaces can thus be introduced as the parameters for delamination prediction as follows

2 1

(a) (b)

Figure 20.Bi-material wedge: with and without delamination

) / ( tan ,

] ) ( )

[( _s^{h 2} _o^{h 2}]]^{1/2 h 1} _s^h _o^h

h //

K^h [([( ))) (( ϕ (129)

) / ( tan ,

] ) ( )

[( _s ² _o ²]]^{1/2 1} _{s o}

v //

K^v [([( ))) (( ϕ (130)

where KK^h, ϕ^h, and K^v,ϕ^v are the combined stress intensity factors and phase angle, and the superscript h and v represent the horizontal and vertical interfaces, respectively. The delamination behaviour along the interface can be characterized by these two parameters completely. The delamination initiates when the stress intensity factor reaches its critical value, i.e., K(ϕ) = Kad(ϕ), where Kad(ϕ) is considered to be a material property as function of phase angle and is termed as the interface adhesion strength for this wedge configuration. Since two bonded interfaces are involved, the delamination will be along the horizontal interface when

) (

) ( ) (

) (

ad ad

v(

v(

h h(

h h(

K K K

K

ϕ (

ϕ ( ϕ

( ϕ

( > (131)

is satisfied. Delamination will occur along the vertical interface if the inequality is reversed.

In document Estrategias Metodológicas Activas que promuevan la Motivación en los Procesos de la Enseñanza y el Aprendizaje de la Lectura y la Escritura (página 56-63)