THE BONDING STRENGTH MODEL
6.2 Diffusion Evolution in Roll Bonding Process
t Al t SST
a r= ⋅r (6.2)
Here rt, Al and rt, SST are the thickness reductions of component layers Al 1100 and SST 304 and they are function of rolling conditions involving the overall thickness reduction, the flow stress of component layers, the friction condition of the component layers’ surfaces and the rollers’ surfaces, the thickness ratio of the component layers, the thickness ratio of the radius to that of stacked metal sheets, as discussed in Chapter IV.
6.2 Diffusion Evolution in Roll Bonding Process
In the previous chapters, the diffusion process at the interface between contacting metals was described. In order to solve the diffusion evolution more efficiently, certain approximations are applied. The diffusion satisfies the diffusion governing Eq. (4.2).
Since the total thickness of the metal sheets is small relative to the width and the length, the following condition is also satisfied:
2
2c 0
z
∂ =
∂ (6.3)
Therefore, the diffusion field can be considered as a 2-D problem.
Figure 6.1 A diffusion state at a bonding interface between Al 1100 and SST 304
Figure 6.1 shows a symmetrical representative volume element for the diffusion between the dissimilar metals. The half plane of the representative volume element is shown in Figure 6.2. The symmetrical plane is defined as x-plane. The y-axis is in the vertical direction and y=0 is the initial contact surface of the extruded Al 1100 and SST 304. The diffusion state is illustrated by the normalized concentration of SST 304-c(x,y,t). SX is the contact length of the Al 1100 and its oxide film, Sdif is the contour length for a particular concentration, λ is the average spacing length of oxide fragments (i.e. the length of the representative volume element in Figure 6.1), δ1 is the distance between the initial contact surface and the Al 1100-alumina interface, w is the width of the crevice between the oxide fragments, ξ is the maximum distance of the contour to the
axis-y (thus χ= ξ-w/2) and η is the maximum height of the contour above the Al 1100-alumina interface. According to the mixture rule, the theoretical peel strength field Gdif(x,y,t) in the diffusion zone can be expressed as:
(
, ,) ( ) (
, ,)
dif Al SST Al
G x y t =G + G −G ⋅c x y t (6.4)
where GAl is the peel strength of Al 1100 and GSST is the peel strength of SST 304.
Figure 6.2 Diffusion front growth in roll bonding of Al 1100 and SST 304
It can be expected, from Eq. (6.4), that the material strength decreases to the strength of Al 1100 at the diffusion front of SST 304 where c(x,y,t)=0. As shown in Figure 6.2, the corresponding contour of the diffusion front has ξ=ξ0 and η=η0 and
Sdif=S0dif. Since the peel strength reaches the minimum value at the diffusion front, it is believed that the peeling path between Al 1100 and SST 304 in peel test will go along the diffusion front as shown by the red dash line in Figure 6.1. Since the alumina film and Al 1100 substrate have relatively sliding by extension in the rolling process, the peeling path would go along the interface (as shown in red solid line) between alumina and Al 1100 besides the diffusion front. This deduction is supported by the observation of the surface of SST 304 after the Al 1100 sheet is peeled off, as discussed in Chapter III.
According to the conservation of energy along peeling path, the overall peel strength G of the representative volume element can be derived:
dif dif X X
Gλ =G S +G S (6.5)
where GX is the peel strength of the interface of Al 1100 and its oxide film.
Reformulating Eq. (6.5), the overall peel strength can be expressed as:
X
dif X
G G f G S
= + λ (6.6)
where the area ratio of the diffusion front surface to the overall Al 1100-SST 304 interface is defined as a function of the diffusion state (involving D0, Q, T and t) and the metal contact area ratio a.
(
, ,)
Sdif
f f a T t
= λ = (6.7)
The function of diffusion front ratio can be determined by the finite element simulation of a 2-D diffusion between Al 1100 and SST 304. Obviously, the function of diffusion front ratio is a monotonically increasing function of the diffusion temperature and time. In the perspective of diffusion front growth, the bonding strength in term of peel strength G will increase as the diffusion temperature or time increases. Meanwhile, the peel strength of Al 1100, GAl, is also changing during the roll bonding process.
Initially, the peel strength of Al 1100 is increased by strain hardening due to the extrusion through the crevice between oxide fragments. Then the annealing effect decreases the peel strength of Al 1100 in the reheating treatment. Considering the strain hardening and annealing effect, the peel strength of Al 1100 is a function of plastic strain εp, heating temperature T and time t.
(
, ,)
Al Al p
G =G ε T t (6.8)
Therefore, the peel strength Eq. (6.6) can be reformulated as
(
, ,) ( , , )
X
dif p X
G G ε T t f a T t G S
= ⋅ + ⋅ λ (6.9)
As shown in Figure 6.2, it could be expected that, when η<0, the peeling failure surface follows the line along the interface between Al 1100 and its oxide film. The
diffusion front curve length Sdif >w only if η>0. Therefore, the diffusion front ratio is derived as following:
( )
, when <0 , , dif , when 0a f a T t S
η λ η
⎧⎪
= ⎨⎪⎩ > (6.10)
Based on the diffusion front growth simulation in Chapter V and the contact area ratio created through a rolling process in Chapter IV, the peel strength for the roll bonding can be estimated by Eq. (6.9) according to the minimum energy principle:
Gtest =MIN G
( )
(6.11)Since the diffusion front profile is in elliptical shape, it is possible that
dif 1 S
λ >
It will lead to
G G> Al (6.12)
In this case, the failure will occur in Al 1100 sheets in a peel test.
As the diffusion continues, the adjacent diffusion zone will merge with each other, as shown in Figure 6.3. The Gtest will be approaching GAl as Sdif approaches λ. Finally, the failure will also occur in Al 1100 sheets in a peel test.
Figure 6.3 Mergence of adjacent diffusion bonding zones