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Wrinkling Phenomena to Explain Vertical Fold Defects in DC-Cast Al-Mg4.5
J. Lee Davis1*, Patricio F. Mendez2 1Novelis Corporation, Molten Metal Process Group, 7421 Pyrite Ct, Castle Rock CO 80108 USA
2Colorado School of Mines, Materials and Metallurgical Engineering Dept, HH 270, Golden CO, 80401 USA
Keywords: oxide, film, vertical fold, can end stock, CES, AA5182, wrinkle, molten aluminum, pseudo-plasticity
Abstract Some aluminum ingots cast by the direct chill method are subject to surface defects on the molten ingot head during casting while others are not. These defects –commonly called “vertical folds” – are frozen into the casting and must be removed prior to rolling. Vertical folds are found on top of the molten ingot surface where areas of thin oxide are (a) bounded by physical constraints and (b) stretched. Physical constraints include (1) substantially thicker oxide or (2) a refractory skim ring adjacent to the thin oxide. The mechanism of wrinkling is suggested for the formation of vertical folds. Wrinkling behavior is described by physical expressions for an elastic sheet in tension whose behavior depends upon thickness h, length L, Young’s modulus E, and Poisson’s ratio υ. The depth and frequency of folds in the thin, elastic sheet parallel to the tensile axis between the two “constraints” can be calculated from these parameters. The observed frequency (and amplitude) of vertical folds in DC-cast aluminum has been found to obey similar wrinkling laws. The frequency-dependence (λ) is examined and found to be related to classic wrinkling parameters but with significant scaling deviations. These deviations may be related to the pseudo-plasticity (self-healing behavior) of the oxide film on the molten surface. A wrinkling model coupled with pseudo-plasticity predicts subtle behaviors in DC casting of Al-Mg4.5 that are not explained by other theories.
Introduction Wrinkling is a natural phenomenon described by physical laws. Fundamental wrinkling behavior for this system is described by mathematical expressions for an elastic sheet in tension which depends on its (a) thickness, (b) length, (c) Young’s modulus, and (d) Poisson’s ratio. The depth and frequency of wrinkles or folds in the elastic sheet parallel to the tensile axis between the two “constraints” can be calculated directly from these parameters. Wrinkling physics are complex. Fundamental mathematics were formulated by (Steigmann 1990). Numerous authors have built upon this foundation with models that vary in their treatment of constraints, in the physical formulation of the system, or in the expected form of mathematical solutions (Chaieb and Melo 2000; Boudaoud and Chaieb 2001; DiDonna 2002; Gioia, DeSimone et al. 2002; Audoly and Boudaoud 2003; Williamson, Wright et al. 2003; Geminard, Bernal et al. 2004; Huang, Hong et al. 2004; Venkataramani 2004; Blair and Kudrolli 2005; Conti, DeSimone et al. 2005; Huang, Hong et al. 2005; Yu, Ye et al. 2005; Rizzieri, Mahadevan et al. 2006; Vliegenthart and Gompper 2006). Cerda and Mahadevan have explored the physics of wrinkling in numerous recent publications: (Cerda and Mahadevan 1998; Cerda, Chaieb et al. 1999; Cerda, Ravi-Chandar et al. 2002; Cerda, Mahadevan et al. 2004; Cerda and Mahadevan 2005). In 2002, they began to examine the wrinkling of a thin, elastic sheet in tension. This was followed by their most important contribution
to this research: “Geometry and physics of wrinkling,” (Cerda and Mahadevan 2003). This work rigorously applies wrinkling theory to a number of systems including thin elastic polyethylene sheets, shrunken apple skins, vesicles, and human skin. It is the thin elastic sheet which draws the attention of the practitioner of the aluminum casting arts, because these stretched sheets look extremely similar to the vertical fold defects found on the molten ingot heads of Al-Mg5 and similar alloys. In early 2006, Mendez suggested that the vertical fold defects in DC ingots were actually wrinkles in a stretched thin oxide film analogous to the descriptions provided by Cerda and Mahadevan. For wrinkling theory as formulated by Cerda and Mahadevan, a relatively small number of physical parameters are required to describe the phenomenon.
FIG. 1. Wrinkles in a polyethylene sheet of length L ≈ 25 cm, width W ≈ 10 cm, and thickness t ≈ 0.01 cm under a uniaxial tensile strain g ≈ 0.10. (Figure courtesy of K. Ravi-Chandar)
Figure 1 Wrinkling of a thin elastic sheet after Cerda
and Mahadevan
When… a thin isotropic elastic sheet of thickness t, width W, length L (t<<W<<L) made of a material with Young’s modulus E and Poisson’s ratio νννν is subject to a longitudinal stretching strain g in its place, it stays flat for g < gc, a critical stretching strain. Further stretching causes the sheet to wrinkle as shown in Figure 1. This nonintuitive behavior arises because the clamped boundaries prevent the sheet from contracting laterally in their vicinity setting up a local biaxial state of stress; i.e. the sheet is sheared near the boundaries. Because of the symmetry of the problem, an element of the sheet near the clamped boundary, but away from its center line, will be unbalanced in the absence of a transverse stress because of the biaxial deformation. This transverse stress is tensile near the clamped boundary and compressive slightly away from it. (Cerda and Mahadevan 2003).
Mathematically, this is defined by:
214
1
2 LT
B
= πλ (1)
* corresponding author: [email protected]
2
ratiosPoisson'υ
thicknessfilmormembraneh
modulussYoung'E
)υ12(1
Ehmembrane elastic of stiffnessbendingB
membrane elastic on field tensionT
wrinklesof length sticcharacteriL
wrinklesof interval or spacing wavelengthλ
2
3
===
−==
===
It is an equation of this form that is found to describe the behavior of vertical folds occurring within the oxide film floating on the molten head of a DC-cast Al-Mg4.5 ingot.
Vertical Folds in Molten Aluminum The vertical folds in DC-cast AA5182 alloy (can-end stock, CES) form on the molten ingot head during DC casting and are visible on the liquid surface in the oxide or skim layer (Figure 2). Vertical folds are usually 1-10 mm in depth or amplitude (A) and run in either continuous bands or periodic swaths (Figure 3) along the surface of an ingot cast by the direct chill (DC) method. The wavelength (λ) or average spacing interval of these folds is generally 20-100 mm as defined in Figure 4.
Figure 2 Vertical folds at the mold/metal interface of CES in a series of oxide breaks; δδδδx is the incremental
length of the oxide break. Novelis Greensboro.
Vertical folds have been observed on aluminum sheet ingots cast by the direct chill method for many years, at least since the 1960s. These defects are generally observed only on aluminum alloys with magnesium content in excess of 2 wt%. These defects are deleterious to the fabrication of rolled can end sheet because the defects must be removed prior to hot-rolling. While almost all aluminum ingots produced by DC-casting are surface-machined (“scalped”) prior to homogenization and hot-rolling, the depth of a typical machining can be as small as 4 mm. The scalping depth might be 10-20 mm to remove the vertical fold defects.
Figure 3 Al-Mg4,5 ingot cast by LHC™ mold with
periodic vertical folds. The depth/amplitude is 2-8mm. Ingot size (x,y,z) 648x1840x5500. Novelis Pinda 2005.
The as-cast surface of these ingots was examined visually and by analytical methods (metallograph, SEM) in order to measure L, n, and w for calculation of lambda (λ). From these measures a comparison was made with elastic wrinkling descriptions.
Experimental Method A complete section of rolling face from a CES ingot (AA5182, Al-Mg4.5) was obtained from Novelis Greensboro via Logan Remelt. The rolling face section was nominally 50 mm thick and 1.5 meters in length. The original ingot was 711x1685x7500 mm and was produced by LHC™ direct chill casting technology. The section was taken 0.9 m from the ingot head, so the corresponding cast length was 5.1-6.6 meters.
Figure 4 Rolling face 1.685m CES ingot from Novelis Greensboro (2006) with definitions of L, w, n, and λλλλ.
Macroscopic vertical folds were measured prior to sectioning, as shown in Figure 4 and 5. The sample was sectioned into smaller pieces and then measured by SEM photomicrographs from an FEI Quanta 600. The measurements were made over a 2 week span. All measurement “set-ups” are shown in Figure 5 and Figure 6. The length scales involved spanned from 100 meters in “macro1” to 10-6 meters in “micro 5”. The ingot surfaces were not treated, nor cleaned. Compressed air was used to blow dust off the
x
y
z
20cm
Rolling face of DC mold
Vertical folds
Oxide breaks
y
z
y
x
z
δx
5 cm
where
3
surface prior to SEM examination. LHC™ ingot surfaces are clean and devoid of surface segregation aside from very small decorative forms, which make the surfaces amenable to direct observation by SEM.
5(a). sample - “Macro 1” measurements (average, n=2)
5(b) Macro 2
Figure 5 Macroscopic measurements made prior to sectioning of the larger sample
6(a) Micro 1
6(b) Micro 2
6(c) Micro 3, 4, and 5 (higher magnification was used for 4&5)
6(d) Micro 6 – note the decorative inverse surface segregation
Figure 6 Microscopic measurements made by SEM
20cm
micro 1
micro 2
micro 3
micro 4
micro 5
macro 2
macro 1(a)
macro 1(b)
micro 6
Backscatter image
y
z
y
z
y
z
y
z
y
z
y
z
4
Results The results from these measurements are found in Table 1 and also in Figure 7.
Table 1. λλλλ and L (bulk) Sample λ (m) L (m)Macro1 3.8E-02 2.8E-01Macro2 7.0E-03 2.5E-02Micro 1 1.0E-03 3.2E-03Micro 2 6.0E-05 7.5E-04Micro 3 6.0E-06 5.0E-05Micro 4 2.5E-06 4.0E-05Micro 5 2.5E-06 2.8E-05Micro 6 1.8E-06 1.0E-05
y = 0.92x + 0.53
R2 = 0.98
-6
-5
-4
-3
-2
-1
0
-7 -6 -5 -4 -3 -2 -1 0
log lambda
log
L
Figure 7 Log-log plot of L versus λλλλ
The mathematical relation between these parameters in Figure 7 can be expressed as:
53.0log92.0log += λL (2)
09.13.0 L=λ (3)
If we are extremely loose with our interpretation of the exponent and – for the sake of elegance – are willing to accept up to 60% error at very small length scales, then: L3
1≈λ (4)
This is a fundamental relationship for data measured across six orders of magnitude on length scale. It is also interesting – but not mathematically relevant – to note that these wrinkles appear to be self-similar across varying scales. It was noted in Equation 1 that the Cerda-Mahadevan relation was defined by:
214
1
2 LT
B
= πλ (1)
or, 21
CL=λ (5) The measurements on the DC-ingot surfaces give a relationship described by Equation 4, where λ = 1/3 L, not λ=CL½. Another interesting result is the apparent variation in wrinkling frequency lambda (λ) with the distance L. Values given in Table 1 are for wrinkles that propagate across the entire distance L – which is essentially the area of “thin” oxide. However, it can be
seen in Figure 3 and Figure 5(a) that lambda is much higher along the leading edge of the vertical fold band. Of course, length L is also smaller and it implies that the oxide is thinner in these areas since bending stiffness B is proportional to the cube of the oxide thickness, h3. This is illustrated in Figure 8.
Figure 8 Close-up of Figure 3
The rough values for λ1 and λ2 versus L from Figure 8 are shown plotted again in Figure 9. From this data, we can see that λ and L are related but not quite in accordance with L3
1≈λ [Eq.4] as λ1
deviates more than other points in the experimental relationship.
lambda2
lambda1
y = 0.92x + 0.53
R2 = 0.98
-6
-5
-4
-3
-2
-1
0
-7 -6 -5 -4 -3 -2 -1 0
log lambda
log
L
Figure 9 Log-log plot of L versus λλλλ, including Figure 8 data
The areas of thicker and thinner oxide were assessed qualitatively by (1) visual observation of the cast in progress i.e. the oxide skin was seen to break and shiny metallic surface appeared as in Figure 2 along with vertical folds within this new area; by (2) SEM/EDS spot probes at various energy values between 5-20 keV; and by (3) examination of the oxide growths on the areas with vertical folds and those areas without – the growth morphologies are complex and with inclusions (salts especially) in areas without folds (Figure 10 and 11). TEM microtomes are being prepared as final
w1
L1
n1
w2
L2
n2
λλλλ1111
λλλλ2222
Thinner oxide
Thicker oxide
Thicker oxide
Thinner oxide
x
y
z
5
confirmation. Areas of oxide within the vertical folds are very clean with a multitude of wrinkles cascading towards smaller and smaller scale as in Figure 12. These clean surfaces likely consist of amorphous oxides and with a few, small inclusions (Figure 13). This is consistent with the classic oxidation literature for molten aluminum as described by (Calvet and Potemkine 1952; Mal'tsev, Chistiakov et al. 1956; Cochran and Sleppy 1961; Thiele 1962; Thiele 1962; Goad 1973; Cochran, Belitskus et al. 1976; Kahl and Fromm 1984; Kahl and Fromm 1984; Kahl and Fromm 1984; Kahl, Yaneva et al. 1984; Kahl and Fromm 1985; Field, Scamans et al. 1987; Stucki, Erbudak et al. 1987; Impey, Stephenson et al. 1988; Silva and Talbot 1989; Impey, Stephenson et al. 1990; Impey 1991; Kaptay 1991). Mal’tsev, Field, and Impey are quite illuminating.
Figure 10 Large inclusion and complex oxide surface
present in "thick oxide." No oriented wrinkling.
Element Line keV KRatio Wt% At% At Prop ChiSquared Al KA1 1.487 0.4831 32.66 23.09 4.3 6.22 Mg KA1 1.254 0.0739 5.32 4.18 0.8 2.08 Ca KA1 3.691 0.0254 1.25 0.59 0.1 1.88 C KA1 0.277 0.0433 14.72 23.38 4.3 4.62 O KA1 0.523 0.2173 36.38 43.38 8.0 2.73 Cl KA1 2.622 0.1221 7.09 3.82 0.7 0.26 Si KA1 1.740 0.0233 1.99 1.35 0.2 11.97 Fe KA1 6.403 0.0117 0.59 0.20 0.0 0.77
Total 1.0000 100.00 100.00 18.4 0.76 Element Line keV KRatio Wt% At% At Prop ChiSquared
Al KA1 1.487 0.5270 37.47 28.20 6.4 1.95 Mg KA1 1.254 0.0828 6.02 5.03 1.1 0.33 Si KA1 1.740 0.0267 2.60 1.88 0.4 7.74 Cl KA1 2.622 0.1723 11.33 6.49 1.5 0.60 Ca KA1 3.691 0.0128 0.71 0.36 0.1 1.81 C KA1 0.277 0.0286 13.01 22.00 5.0 1.89 O KA1 0.523 0.1368 27.96 35.48 8.0 2.24 S KA1 2.307 0.0131 0.90 0.57 0.1 0.50
Total 1.0000 100.00 100.00 22.5 0.47 Figure 11 EDS spectra – large inclusion mass at 20keV
Recently this area was re-examined by a number of authors (Campbell, Kalia et al. 1999; Tsunekawa, Tani et al. 2001; Akagwu, Brooks et al. 2003; Akagwu, Brooks et al. 2003; Bainbridge, Taylor et al. 2004; De, Mukhopadhyay et al. 2004; Doremus 2004; Campbell, Aral et al. 2005; DeYoung, McGinnis et al. 2005; Kolek, DeYoung et al. 2005; Giuranno, Ricci et al. 2006; Shih and Liu 2006; Syvertsen 2006; Zhang, Zhang et al. 2006) with little change in the formulation for oxidation process and products.
Figure 12 Clean, amorphous oxide surface with
cascading wrinkles adjacent to a large macroscopic vertical fold in thin oxide region.
Figure 13 Small inclusion imbedded in the amorphous
"thin oxide" near a macroscopic vertical fold (a) Gray bulk – EDS shows Ca, C, O major and (Cl) minor (b) White block – EDS shows Fe, O, and (Cl, C) (c) White sphere – EDS shows Fe, Si, O and (Ca, C) (d) Light gray prism – EDS shows Si, Mg, Ca, O
Literature values for oxide thickness on molten Al alloys are 10 nm (Impey 1991) for a newly oxidized melt (<1 min exposure) at 700°C while they are 400 nm (Akagwu, Brooks et al. 2003; More, Tortorelli et al. 2003) on an old melt with exposure times >1 hour and T>750°C. At large thicknesses, the oxide starts to form “dross” with layers of both metallic aluminum and oxide imbedded together. It is no longer a relatively simple oxide mass. (Newkirk and Dizio 1987; Impey, Stephenson et al. 1988; Impey, Stephenson et al. 1990; Impey 1991; Weirauch, Balaba et al. 1995; Roy and Sahai 1997) This is due to interaction of fractures in the newly
(a)
(c) (b)
(d)
Vertical fold (macro-scale)
Backscatter image
y
z
y
z
y
z
Vertical folds (micro-scales)
6
crystallizing oxide film and the surface tension of the liquid metal against it. This produces a capillary effect that layers the material in oxide/metal/oxide configurations, especially in Al-Mg alloys. Taken as a whole, this indicates that wrinkling occurs in the new, thinner oxide and not in the old, thicker material. This leads to a hypothesis where “wrinkles occur in films that are thinner relative to the films immediately bound to them.” Key discussions investigating this hypothesis include:
1. Scaling laws for wrinkling frequency lambda (λ) versus wrinkling length (L).
2. Forces within the DC-casting system that could produce tension fields on the solid oxide film that are capable of forming wrinkles.
3. Complexities in the oxidation process and its self-healing nature that might lead to deviations in the scaling law.
Discussion
If one compares Cerda/Mahadevan’s formulation for wrinkling in thin elastic films (Equation 5) to the relationship obtained in this work (Equation 4), there is a significant discrepancy.
Actual Results L31≈λ (4)
Classic Theory 21
CL=λ (5)
For these equations to be consistent over the range of measures in this study, CL½
Cerda and 0.33LDavis must be equivalent. This is not likely. If we consider the entire description of lambda, λ and L (Eq.1) and note there are specific processes that occur due solely to (a) semi-continuous casting of molten metal and (b) oxidation of such molten metal; and transform the governing equation by these processes; then perhaps an explanation for the discrepancy can be found.
Figure 14 Idealized DC ingot head (molten) with uniform solid oxide breaks of δδδδx and velocities
Vcasting ≅≅≅≅ Vcorner >> Vend >> Vlong face. In DC casting, the length of molten metal surface exposure to atmosphere is governed by two factors (1) the casting speed (~1 mm/s) and (2) the relative speed of the oxide skin on top of the molten metal. When the oxide skin travels at exactly the same speed as the solid ingot, then we observe that the oxide undergoes continual fracturing into small strips that are δx wide (Figure 2) and about the length of the nearest mold wall. From data in this study, we see δx as small as ~10 microns in “micro 6” (upper left in Fig. 6.d photomicrograph). The mold corners insure that the oxide strips are broken into four separate regions (1-4) bounded by the mold wall, the mold corners, and the distribution system as shown in Figure 14 (assuming there is no skim ring). The rupture frequency should be dictated by the fracture behavior of the
system, which is quite complex. An indication of the behavior may be obtained by examining Young’s modulus for the oxide, since it is involved in (a) classic elastic behavior, (b) wrinkling theory, (c) pseudo-plasticity and (c) fracture mechanics. Elastic solid mechanics If we consider classic elastic behavior for brittle materials, the casting speed acts as the machine strain rate for the oxide skin. The non-trivial question is “what is the effective gage length(s)?” For a sheet of homogenous oxide in a 711x1690 mm ingot head, there is a dimension of x≈300mm from the distribution system to the rolling face (Figure 2). This gives a strain rate of (103ms-1 /0.3m) = 3000s-1 – relatively high. Based on stress-strain data from (Field, Scamans et al. 1987) in Figure 15, the sheet could only stretch a small and finite length before its maximum strain value would be exceeded. Young’s modulus calculated by (Field, et.al. 1987) for a 300 nm amorphous film on a solid substrate was 55+/-3 GPa with water vapor present and 213+/-69 GPa without water vapor. Literature values for Young’s modulus for Al2O3 and other spinel-type structures – various forms of crystalline corundum, magnesium-aluminate spinel, and related oxides at room temperature – range from 300 to 400 GPa.
0
50
100
150
200
250
300
0.000 0.001 0.002 0.003 0.004
elongation (in/in)
Str
eng
th (
MP
a)
w/out water vaporw/water vapor
Figure 15 Idealized stress-strain relationship for aluminum oxide in Al-Mg4.2 after Field, et.al.
The total elastic length l for the oxide (w/ water vapor) would be:
mmmmLl mmmm 06.1)00354.0)(300( === ε
As an aside, water vapor is impossible to avoid in DC casting at ambient atmosphere and has recently been added intentionally by some producers under patent (Kolek, DeYoung et al. 2005). Its beneficial properties for creating and stabilizing protective oxides during casting have long been known (Gauthier 1938; Whitaker and Heath 1953; Mal'tsev, Chistiakov et al. 1956). Thus, if the oxide were an ideal sheet of length 300mm and thickness (h = 10-8 to 10-7 m) there would be ~1 mm extension of the plate prior to rupture. At a typical casting speed of 1 mm/s, this means that the rupture frequency would be ~1 Hz. When the oxide is moving uniformly during DC casting, we observe this to be the case qualitatively. No measurements have been performed on real-time rupture frequency of the sheet during DC-casting. Wrinkling Wrinkling theory describes the frequency lambda, λ, as directly proportional to E¼ and film thickness h¾. However, we know that (1) film thickness is increasing constantly due to oxidation and (2) it is likely that the effective modulus changes with thickness due
Mini-bag Vend
Vcorner
Vlong face
y
x
E=213 GPa
E=55 GPa
εσ=E
1
2
3
4
7
to defects within the film. For insight into these related phenomena, we examine both the classic oxide rupture work on liquid aluminium surfaces by (Kahl and Fromm 1984) and (Syvertsen 2006) and limited data on the variability of elastic modulus in dynamic oxide films. Eventually, this will lead to an examination of the dynamic rupture of oxide films on aluminum melts, the interaction between rupture and oxide growth (film thickening), and to a qualitative exploration of pseudo-plasticity. Young’s modulus It is generally accepted that materials of a given class – aluminum alloys, steels, enameling irons, beryllium-coppers – have a “single” or generally accepted modulus of elasticity. However, thin films are a unique material unto themselves with properties that often depend more on substrate than the material itself. Further complicating matters is the oxidation process, which tends to change the long-range structural integrity of the film over time and with thickness. The best data in our specific area comes again from (Field, Scamans et al. 1987) where they have two films of anodic Al2O3 on a solid substrate but two entirely different moduli (Figure 15). It could be argued that these are not the same material: one is hydroxylated while the other is not. Conversely, it could also be argued that they are the same material but the overall bulk structure is sufficiently different to produce changes in the bulk modulus. Most of the structural spinel oxides: (Me+2)(Me+3)2O4; including MgAl 2O4, have a very similar range of moduli – about 270-310 GPa, owing to their unique structural configuration (Mitchell 1999). Aluminum oxide (α–corundum) has bulk values for E of 400-420 GPa (Wachtman 1996). However, researchers report values for Young’s modulus for oxides – which have significant disorder – that are dependent on density/order/defect-size/porosity (which is proportional to thickness) and temperature. Standard equations for elastic modulus variation from the standard EPRI formulation (Armitt, Holmes et al. 1978) include:
)1(
)21(
1
12
pEE
NcEE
dT
dE
E
T
T
TEE
p
N
o
M
Mo
−=
+=
⋅⋅
+=
−π
where Eo= the bulk modulus at T=0K N = no of defects of average size c p = porosity
Note that the effective modulus used for fracture mechanical calculations – which is an indication of the fracture toughness KIc
– is dependent on the number of defects (N) and the square of their average size. E also depends on the porosity (p) of the material directly by (1-p). (Wachtman, Tefft et al. 1961) proposed that:
−−=
T
TbTEE o
o exp
where b and To are empirical constants and To = the Debye temperature (Mitchell 1999) summarizes the temperature dependence of E for MgO, Al2O3, and MgAl2O4 from other authors (Goto, Anderson et al. 1989) in Figure 16. These results are confirmed by (Isaak,
Anderson et al. 1989) and show a decrease in E with temperature from a purely theoretical basis. Thus, the temperature dependence is decoupled from scale growth, oxide rupture, etc. which is the basis of the more empirical approach described by Wachtman and (Armitt, Holmes et al.) in the EPRI formulations. (McCartney 2005) gives the modulus temperature relation for MgAl2O4 from ambient to 600°C as E = Ebulk – 0.05∆T.
Figure 16 summary of theoretical temperature
dependence of elastic constants for various crystal orientations in MgO, Al2O3, and MgAl2O4
From both theoretical and experimental bases, Young’s modulus for an aluminum oxide could decrease some significant proportion – even an order of magnitude – from room temperature to 1000K. This is due to the temperature dependence of the fundamental oscillations of atomic bonds in lattice space as well as bulk structure changes (porosity, defect creation) associated with the increased temperature. For a treatment of the change in oxide thickness with time, we start again with research from (Kahl and Fromm 1984) on the dynamic rupture of oxide films on molten aluminum alloys. Then we will proceed through the oxidation literature and link the dynamic rupture and oxidation behavior to a process described by Schutze as pseudo-plasticity. Dynamic oxide film rupture on liquid aluminum (Kahl and Fromm 1984) obtained values for the torque required to break the oxide film (in-situ on liquid aluminum alloys) versus distance, as shown in Figure 17. They found the basic behavior of the system was not governed by the average oxide thickness, as expected. Instead, it was governed by a factor over which they had no control: the newly formed oxide surface. They refer to this as “the young skin.” This is oxide that forms rapidly upon rupture of the older, thicker oxide film. It is the “self-repairing” part of the aluminum oxide film.
Another indication for the significant role of the young skin formed between cracks is the fact, that the force measured during cracking is reduced only by about 20 %. i.e., a relatively strong new oxide scale must have been formed between the crack edges with a rate comparable to the propagation
(6)
(7)
(8)
(9)
8
rate of the crack. Otherwise, the crack peaks in fig. 4 would drop to zero.
(Kahl and Fromm 1984)
Fig.4. Typical curve force f vs. time t(distance) measured with apparatus L for linear die movement. (1) die above the melt; (2) die indentation; (3) oxidation period; (4) skin stressed; (5) crack formation; (6) die movement stopped; (7) skin destroyed behind die; (8) skin destroyed between die [1] and [2]; and (9) dies removed from the melt.
Figure 17 Kahl and Fromm (1984) linear (L) device for measurement of oxide “strength” on liquid Al
This means that once a rupture in the oxide film starts, subsequent breakage is observed in the same location. Kahl and Fromm note only a 20-40% reduction in the required force to perpetuate oxide rupture after the initial failure. This is important for two reasons. First, observations in this study show that wrinkling occurs – macroscopic and microscopic – in thinner oxide breaks, not in thicker oxide plates. Kahl and Fromm’s sketches from 1984 show wrinkling patterns almost identical to those observed in this study. Next, it provides insight into the high dynamic “strength” of the oxide as it ruptures and re-oxidizes.
Fig.15a to d. Typical test curves for four different liquids. a) Aluminium, selfhealing skin, η=1.1cP; b) lead, selfhealing skin; c) glycerol, no skin η=1480 cP; d) milk, skin not selfhealing
Figure 18 Kahl and Fromm measurements for self-healing and not self-healing films – rotational apparatus
Kahl and Fromm report that the torque required to tear an aluminum oxide film is almost constant even after the oxide is ruptured (Figure 18). This would be analogous to tearing a sheet of paper in half and having the paper “resist” long after the two halves have been separated. Kahl and Fromm reported torque values of 0.003-0.006 Nm for the various oxide films studied and tension fields of 1.5-2.5 N/m for their linear apparatus. The surface tension of liquid aluminum with an oxidized surface is ~0.86 N/m (Kalazhokov, Kalazhokov et al. 2003).
So, the rupture behavior of the overall oxide mass has been observed to be controlled by the previous rupture site. Essentially, the ideal oxide mass ruptures and creates surface with oxide extension, translation, and re-oxidation in the same location repeatedly. In the rupture site, new oxide will be 10-20 nm thick and amorphous according to (Impey, Stephenson et al. 1988; Impey, Stephenson et al. 1990; Impey 1991). If we take the smallest δx = 10 microns (Fig 6.d. top left), then the crosshead (cast) speed would imply that the smallest oxide plate could be δx/ε or approximately 10mm (10-2m).
Oxidation behavior We know from published research that the oxidation behavior for molten aluminum – with or without magnesium – is complicated. And from images of the ingots and data collected at this casting pit, we see a number of distinct oxide releases occur over this time period. The size of these macroscopic releases is much larger than the 10mm (10-2m) ideally calculated above. The time period involved in oxidation on the molten head of a DC-ingot is [cast length]/[cast speed] which is on the order of 102 minutes. From Figure 19, it can be seen that there are nominally 10-20 oxide releases per ingot per cast. Also, the frequency of releases is not consistent (note the large gap between releases 11 and 12). Despite that, the average dwell time for oxide on the surface of the ingot head is 102 min /15 = 6-8 min. This gives an average macroscopic oxide length of 7500mm/15 = 500mm: significantly larger than the 10mm calculated value. Note, this is the “inverse” length IL of the length L in Figure 4 or 8, where L and IL are related by niΣ L + niΣ IL = cast length (n=count; L, IL).
Figure 19 Novelis Greensboro 711x1690x7500mm Al-
Mg5 ingots w/periodic oxide breaks (n=13) (dark areas)
This behavior must be linked with the oxidation processes of the aluminum melt. When the oxide has ruptured and instantaneously recreated a new film 5-10 nm thick, this new oxide and its older brethren grow (thicken) by some oxidation law. Oxidation studies for pure liquid aluminum and Al-Mg4/Mg5 are common, but most of studies deal with dross formation and time intervals on the order of hours. Weight gain studies on a time scale of seconds are required and are not typically found in literature, whose main focus has been reduction of dross generation – not elimination of vertical folds. However, some pertinent data for short-exposure-time aluminum oxidation exist. (Cochran and Sleppy 1961) describe the oxidation of Al and Al-Mg2.5 at 550°C as “more nearly parabolic than linear.” Research by (Field, Scamans et al. 1987) follows up
Crack peaks
1 2 3
4 5
6 7
8 9 10 11
12 13
9
by describing the high-temperature solid oxidation process for pure Al in three stages:
1. an initial logarithmic rise due to thickening of the original overlying amorphous film of γ-Al 2O3
2. an adjacent, S-shaped branch as a result of crystalline film formation at the oxide/metal interface, and
3. a final linear increase in weight due to further nucleation and growth of crystalline oxides
Work by (Silva and Talbot 1989) as well as (Impey, Stephenson et al. 1988; Impey, Stephenson et al. 1990) in liquid aluminum of both 99.99Al and Al-Mg5 compositions support this mechanism for the molten state as seen in Figure 20. Using Silva and Talbot’s data to assess the oxide growth during the ~8 minute timeframe observed we obtain an oxide thickness (h) as follows:
h2 = k1t and kp=k1½ (10)
Al-Mg5
0.0
0.2
0.4
0.6
0.8
0.0 0.5 1.0 1.5 2.0Time (h)
mas
s ga
in/ m
ax m
ass
gain
(mg
cm-2
)
725°C
700°C
675°C
650°C
Figure 20 Silva and Talbot weight gain data
for molten Al-Mg5 oxidation at 700°C
If the parabolic oxidation constant is estimated from Silva’s data at kp=9 x10-9 cm2/s, then the oxide growth in 480s is 208 nm – a reasonable estimate based on literature values. This represents an order of magnitude increase from 20 nm to 228 nm thickness. If growth is linear where h=k1t over the first 5-10 minutes with k1=1x10-9 m/s, the thickness would be h=480nm in t=480s. If we consider that the molten ingot head is ~300mm wide, it is possible for an incremental oxide of thickness δx to stay on the molten head for 500mm + 300mm = 800mm at 60mm/min cast speed for a dwell time of t=800s. This gives h=268nm (parabolic) to h=800 nm (linear). This provides a rough, semi-quantitative description of the oxide thickness over the cast length and across the molten ingot head. If we assume that the thin oxide areas move at approximately the casting speed, then there will be a gradual increase in thickness of the “thin film” as it approaches the “thick film.” This interface will be thicker at one end but discontinuous at both as shown in Figure 21. This is the root cause for the differences in λ1 and λ2 in Figure 8. This infers that thin oxide areas tend to keep tearing and healing – with little strength loss – through a balance between the rupture and oxidation processes. It also suggests the oxide should be thinnest along the edge that is continually breaking (Area B in Fig 21).
(a) time t=t1
(b) time t2>t1
A – no wrinkles – thick oxide acts as a “clamp”
B – microscopic wrinkles in small breaks shown as ||| (δx) C – small macroscopic wrinkles decaying from A D – wrinkles across entire length scale E – small macro wrinkles decaying from F but less than C F – no wrinkles – thick oxide acts as a “clamp”
Figure 21 Oxide thickness profile (idealized) The possibility of the aluminum oxide’s self-healing process somehow balancing out with its rupture process leads to a discussion of pseudo-plasticity.
Pseudo-plasticity From SEM examinations and from direct observation of the DC casting process itself, it is likely that ruptures within the oxide film occur at everywhere between 20 microns < L < 0.5 meters, producing the complex and cascading wrinkling behavior seen in this paper. What is interesting to an observer during the DC-cast is how the oxide appears to be alternately (1) moving with the cast speed – which we designate as “transitory” – or (2) not moving on the molten head despite the cast speed, “stationary.” When the oxide is transitory, it is often very bright and shiny in appearance. SEM verifies the oxide is very thin. It is probably amorphous based on past research. To the naked eye, it appears that the oxide is actually stretching on the surface of the liquid metal as it moves. From mechanical testing of oxides, a pertinent cross-section of which is reproduced in Table 2, we know this is not possible. What perpetuates this illusion of a plastic oxide film stretching across the molten surface, broken only by the geometrical constraint of the rectangular corner of the mold?
k2 = 9x10-9cm2/s (t<0.5hr @ 700°C)
A B
C
D E
F
Not to Scale Side View
Not to Scale Side View
Oxide 1 Oxide 2
Oxide 1
Oxide 2 Edge of distributor/mini-bag R
10
Table 2. Mechanical properties for aluminum oxide films Author Impey Akagwu
Film structure
amorphous anodic g-
w/out water vapor
amorphous anodic g- w/water vapor
amorphous crystalline amorphous crystalline
Thickness (nm)
300 300 10 400 ~20 400
Substrate solid Al solid Al liquid Al liquid Al liquid Al liquid Al
Temp (C) 25 25 700 950 700 700
E (GPa) 213 +/-69 54.7 +/-2.7 n/a n/a n/a n/a
sfracture
(MPa)252 175 147 55 100 5
elongation (in/in)
1.52E-03 3.54E-03 n/a n/a n/a n/a
Field, et.al. Kahl, Fromm
The answer may well be pseudo-plasticity as defined for oxides by (Schutze 1990). Schutze and subsequent authors describe some damage-resistant oxide scales on solid substrates at high temperatures as having the property of pseudo-plasticity. The basic premise for pseudo-plasticity is that the oxidation rate heals the newly forming cracks at the same rate that they are forming. If the strengths of the old and new films are similar, then the oxide undergoes an apparent plastic deformation well in excess of its tensile elongation. Schutze’s formulation (Eq 11) provides the critical strain rate below which the oxide film will undergo sufficiently balanced oxidation in order to maintain a protective film while undergoing tensile strain, e.g. pseudo-plasticity. This formulation was made originally made for films in boiler tubes with thickness range of 1<h<20 microns, not 10-100 nanometers. At such thicknesses, the values for ε-dotcrit are in the range of 10-9
to 10-6/s.
1
1
12 +
=
mmp
critical hlk εε
&&
(11)
where h = oxide film thickness l = observed distance between scale cracks at a strain rate of ε-dot1, m = experimental dependence of l on ε-dot kp = parabolic oxidation constant
Schutze’s expression evaluates ε-dotcrit for aluminum oxides in the DC-casting system according to Figure 22. With assumptions for exponent m, it can be seen that pseudo-plastic behavior in the aluminum oxide system is possible. Pseudo-plasticity would occur in Figure 21 at the location shown as “oxide breaking here.” This might also account for the observation where the oxide film changes from “transitory” to “stationary” behavior after a time delay of 6-8 minutes (once it achieves an overall thickness in the range of 102-103 nm) as shown in Figure 19 and 21. To the authors’ knowledge, this is the first testable hypothesis suggested for the behavior of the oxide skin on the molten head of a DC ingot. Additional research is required to confirm the details (local strain rate ε-dot, length L, pseudo-plasticity exponent m) of the system and its application to thin oxides on molten aluminum. However, preliminary data presented in this study are compelling. If the overall oxide thickness increases to a point where h exceeds hcrit, then Schutze predicts that pseudo-plasticity will cease and
brittle oxide fracture will occur. Critical thickness values from Figure 22 are 20 nm < hcrit < 600 nm for exponents 0.5<m<1.0.
1
10
100
1,000
10,000
100,000
1E-0
9
1E-0
8
1E-0
7
1E-0
6
1E-0
5
1E-0
4
1E-0
3
film thickness, h (m)
criti
cal s
trai
n ra
te (
Hz)
.
m=0.5
m=2.0
m=1.0brittle
Generalizedstrain rate forDC casting
ε-dot ~1000/s
pseudo-plastic
Figure 22 Schutze calculation for critical strain rate below which pseudo-plasticity should occur within aluminum oxide films during direct-chill casting
Fracture mechanics The failure of an oxide film is a complex process governed by fracture mechanics. While the equations for fracture are straightforward (Equation 12-13), they are complicated by (1) non-axisymmetric stress application, (2) unknown defect sizes and distribution, (3) lack of precise data for Young’s modulus, and (4) the process of pseudo-plasticity (Schutze 1990). Brittle fracture conditions are generally given by:
Critical fracture stress 2
1
)( c
K Icc π
σ = (12)
Critical fracture strain 2
1
)( cEF
K Icc π
ε = (13)
where KIc = Type I fracture toughness ≅ (2γiE)½
c = defect size E = Young’s modulus F = geometry factor = 1 for imbedded defects γi = intrinsic surface energy
(Field, Scamans et al. 1987) presented a rupture description for a perturbation in an oxide over-layer. They calculated the size of a crystalline particle at the metal-substrate/oxide interface required to fracture an amorphous over-layer. (Impey 1991) used this exact description without reference to Field and Scamans et al.
Critical fracture thickness 3
1
2
2
12
=
φγ
M
R
Eh i
crit (14)
where R = radius of interface curvature M = scale displacement vector for perturbation φ = Pilling-Bedworth ratio
Instead, (Impey 1991) and (Akagwu, Brooks et al. 2003) calculated the fracture stress of oxide films by insertion of a glass probe into the oxidized melt according to Eq 15.
kp = 10-10 m2/s ε-dot1 = 103/s l = 10-5m
10nm 100nm 1um 10um 100um
11
grxxx
rxxrhF Lox
+−++−−= )(
232
)(2 223
21
πππρσπ
where F1 = force of the probe at fracture of oxide r = radius of the probe h = oxide thickness σox = fracture stress of the oxide ρL = density of the liquid x = depth of probe below surface at fracture g = gravitational constant
Assuming that defect size is on the same order as the film thickness, e.g. c~h, values from Table 2 can be used to estimate KIc from Eq 12. The fracture strain can be estimated from Eq 13. This gives KIc ≅ 0.03 MPa√m, three orders magnitude lower than published values for alpha-Al2O3 (Schutze, 2005). To obtain fracture toughness values similar to those for alpha alumina, the critical equivalent defect size (c) would have to be on the order of 1000 times larger than the oxide thickness, h. The equivalent fracture strain εf ≅ 1x10-8 with KIc estimates the intrinsic surface energy at γi ≅ 10-3 J/m2. This is extremely low compared to the published values in the range of 1-10 J/m2 and small compared to the surface tension of oxidized liquid aluminum (~1 J/m2). (Kahl and Fromm 1984) calculated a fracture strength of σc=100 MPa for a new 20 nm oxide skin and σc=5 MPa for an older 400 nm film. (Impey 1991) calculated a rupture strength of σc=147 MPa for a 10 nm amorphous film at 700°C while (Akagwu, Brooks et al. 2003) calculated a value of σc=55 MPa for a 400nm α-Al 2O3 film at 950°C (Table 2). Literature values for the fracture strength of aluminum oxide – typically crystalline α-Al 2O3 – are σc=200-300MPa. While not conclusive, it appears that fracture mechanics alone do not account for the unexpectedly high fracture strength of the film. Instead, another process – pseudo-plasticity – controls the overall behavior of the film until a critical oxide thickness is reached, when pseudo-plasticity “fails” and brittle fracture ensues.
To summarize: 1. The oxide ruptures across lengths from 20 microns to
500 mm. An “ideal” oxide plate would have L~10-2m. 2. Scale lengths are commingled. This may be dependent
upon local fracture mechanics 3. The oxide thickness varies from ~ 10-400 nm. 4. The mechanical properties of the oxide deteriorate with
increasing thickness; i.e. E is proportional to h-1. 5. Pseudo-plasticity plays a large role in the oxidation
behavior of the molten surface during casting. It may be responsible for transitory to stationary behavior.
6. A graphical representation of the oxide structure can be found in Figure 21.
Reconciliation of λλλλ versus L scaling
From the preceding discussion, it is possible to re-evaluate Equations 4 and 5 in light of specific behavior in the DC casting system. Starting with Equation 1, Cerda/Mahadevan’s most generally accessible formulation for wrinkling, we find:
214
1
2 LT
B
= πλ (1)
)1(12
3
υ−= Eh
B (16)
but h and t are related by:
tkh 2= (17)
Assuming that the average velocity is applicable to an average incremental oxide area yδx, then time t and characteristic length L are related by the casting speed (V) where:
1−= LVt (18)
Substituting for h and then t, we obtain Equation 19.
87
41
23
23
27
27
23
23
23
32
32
244
32
244
232244
232244
2244
)1(12
2
)1(12
2
)1(122
)1(12
)(2
)1(122
2
LTv
Ek
Tv
ELk
LvT
Ek
LT
vLEk
LT
tEk
LT
B
−=
−=
−=
−=
−=
=
υπλ
υπλ
υπλ
υπλ
υπλ
πλ
Values for E have been shown to be in the range 50< E <200 GPa. The tension field, T should be on the order of the surface tension or the intrinsic oxide strength, e.g. 0<T<10 N/m (J/m2) – or else the oxide would rupture in a brittle manner without pseudo-plasticity. Poisson’s ratio υ~0.2 and the parabolic rate constant is on the order of kp = 9 x10-9 cm2/s. This generates the relationship of Equation 20. If the oxidation process is assumed to be linear, where h=k1t, and k1 ~ 1x10-9m/s, then Equation 21 can be derived. Neither expression describes the experimental data as embodied in Equation 3.
Parabolic 876103 Lx −=λ (20)
Linear 125.1673L=λ (21)
Actual 09.13.0 L=λ (3)
The role of pseudo-plasticity has not been treated mathematically here for reasons of space and complexity. However, one notes that oxide thickness h features prominently in pseudo-plasticity and stiffness B. Substitutions of pseudo-plasticity formulations into the scaling laws can be done once the empirical exponent m is known. Also, the role of crystalline oxide defects at the melt/oxide interface (as described by (Field, Scamans et al. 1987) in the fracture mechanics of the system has not been evaluated but may be important as suggested by discrepancy between theoretical and experimental results for fracture strength σc. These mismatches (and others) in theoretical and experimental observations require additional research.
Summarizing, one notes that the observed scaling exponent of 1.09 falls between linear and parabolic oxidation assumptions. It is difficult to resist a hypothesis where “mixed” oxidation modes “smooth out” the exponent to a value somewhere between the theoretical limits for the two treatments. This is qualitatively in agreement with (Field, Scamans et al. 1987).
(19)
(15)
12
Forces to Generate Wrinkles The requirements to generate a wrinkle in an oxide are:
1. Force sufficient to exceed the threshold value for the tension field, T.
2. Force that is less than the rupture strength of the oxide.
We assume that the tension field T will be locally continuous across adjacent areas of thick and thin oxide and that it will be uniform within a small local area defined by∫
xxy
δ
0
.
There are a number of candidate forces that may act upon the head of the ingot and its oxide, including:
1. Fluid flow (fluid shear at the liquid / oxide interface) 2. Casting speed (discussed in earlier section) 3. Machine vibration (from hydraulic cylinder, etc.) 4. Unknown forces
For the candidate force to cause the wrinkling, the force must be present and applied parallel to the ultimate location of the wrinkles. On logical grounds alone, machine vibration can be eliminated. Although machine vibration is present in all systems, the force cannot be interpreted as aligned in any geometric sense that is centered on the ingot head. Machine vibration may make for energy spikes that lead to oxide failure, but not to wrinkling. Drag Forces from Fluid Flow Fluid flow in the ingot head is a candidate force. The author and colleagues (Grealy, Davis et al. 2000; Grealy, Davis et al. 2000; Tondel, Grealy et al. 2000) and many others have shown that the sub-surface flow velocities in DC castings are on the order of 0.01-1.0 m/s. They also have an interesting characteristic shape that looks very similar to the shape of the periodic vertical folds found in many LHC™ Al-Mg5 ingots. Compare the flow field in Figure 23 to the shape of the oxide releases in Fig 2 through Fig 5.
Figure 23 Fluid velocity vectors (mm/s) 20mm beneath
the melt surface from (Grealy, Davis et al. 2000)
For external flow, the Reynolds number is given by:
υVL
x =Re (22)
where V = velocity, x = distance along length L, and ν = absolute viscosity.
And the thickness of the laminar boundary layer, δ is given by:
2
1
Re
0.5
xx=δ (23)
In such an instance, it can be shown that the drag force, FD arising on an object from fluid shear can be calculated from:
21
23
21
21
664.0
00
xVbF
dxbdAF
D
xx
D
µρ
ττ
=
== ∫∫
where A = area of free flow, b = width of free boundary, τ = shear stress on A, ρ = fluid density, µ = kinematic viscosity.
From the force FD, the tension field T can be calculated by: T = FD/x (25)
This leads to a general relationship between T and distance from the centerline of the ingot (x) that is shown in Figure 24.
y = 0.028x-0.500
y = 0.080x-0.500y = 0.146x-0.500
y = 0.225x-0.500
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0.00 0.05 0.10 0.15 0.20 0.25
Distance x (m)
T, T
ensi
on F
ield
(N/m
)0.2 0.4 0.6 0.8
Fluid Velocity (m/s)
thin oxide ruptures
thin oxide wrinkles
Figure 24 Relationship between T and x for fluid drag
along the molten metal/oxide interface
This relationship can be generalized by numerical methods to a velocity distribution with respect to distance x along characteristic length, L. In this case, we see that 0.01 < FD < 0.10 N and that 0.1 < T < 3.2 N/m for 0 < x < L as described in Figure 25.
A number of interesting observations arise from this analysis. First, note that the tension field is not everywhere constant because the fluid drag develops higher force as x�0 along the external flow path from x�L. The calculated values for this tension field are in excess of the surface tension of oxidized liquid aluminum (~0.86 N/m). This is in keeping with observations made in various casting centers where the free fluid surface is observed to be disturbed close to the centerline of the ingot during DC casting. This is also seen in Figure 2 where the oxide break is seen to occur near the centerline of the ingot during casting.
From Equation 1 and the constants developed earlier for E and L, we can estimate the tension fields required to wrinkle oxides of 10nm and 100nm. We find the requirements for oxide rupture are not met by the fluid drag but they are sufficient for wrinkling. The rupture tension of the lightly oxidized melt – the thin oxide – is given by its apparent surface tension, a value of ~ 0.86 N/m.
y
x
x,L
(24)
13
Thus, TWRINKLE, t=10nm ≅ 10-5 N/m
TWRINKLE, t=100nm ≅ 0.1 N/m TRUPTURE > 0.86 N/m TFLUID < 1.0 N/m
So, it is possible that some of the wrinkling behavior is brought about by fluid drag as described in Figure 25.
0.01
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.2
0.4
0.6
0.8
1.0
Velocity V, m/s
Length x, m
Boundary Layer
δ (delta), m0.000-0.0060.001 incre
0.001
0.002
0.003
0.01
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.2
0.4
0.6
0.8
1.0
Velocity V, m/s
Length x, m
Fluid Drag on Plate (shear) N
FD, N0.00-0.150.01 incre
0.01
0.020.03
0.01
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.2
0.4
0.6
0.8
1.0
Velocity V, m/s
Length x, m
Tension Field, T (N/m)
T, N/m0.0-3.20.1 incre
0.1
0.20.3
0.86
Figure 25 Boundary layer δδδδ, drag force FD, and
tension field T versus length x and fluid velocity V
The values for tension field, T that generate observed values (~1/3) for the pre-L constant in Eq. 3 are in the range of 10-6-10-5 N/m if linear oxidation behavior is assumed. This is the tension field, T, required to wrinkle a film 10-20 nm thick. This supports the possibility that tiny 10-20 micron-length non-axisymmetric features perpendicular to the z-axis in Figure 6(d) are produced by fluid drag. It is also compelling that the rupture tension of the oxide should be exceeded - by fluid drag - near the exit of the
metal distribution system (near the center-line of the ingot). Preferential ruptures at this location are observed in DC casting, but no mechanism for such behavior had been suggested except for fluid turbulence and its corresponding increase in the vertical translation vector (M in Eq 14) of the oxide film.
It is difficult to imagine, however, a large axi-symmetric fluid drag stress on the oxide film given the complexity of the fluid field. Only a large, symmetric stress could create the depth of wrinkling observed on a macroscopic scale (1-10 millimeters). Computational fluid dynamics (CFD) would have to be employed to examine this in detail, then coupled with the governing film wrinkling, growth, and fracture equations to see if these macroscopic folds could indeed arise from fluid drag. The more plausible force available to produce large wrinkles on the scale observed with the naked eye is the casting speed.
Forces from Casting Speed The cross-head velocity associated with the casting speed has been discussed in previous sections. We observe that vertical folds occur with a small-angle deviation from the axial loading provided by the casting speed. We also observe that the oxide floating on the ingot head is intimately connected to the oxide that is attached to the solid substrate of the ingot below the mold by the process of pseudo-plasticity. Translation of the oxide through 90 degrees at the meniscus radius (R in Figure 21) is also trivial when compared to oxide film thickness. The simplest definition for curvature, κ is h/R. In this case, κ=10-8m/10-3m; close to zero. The oxide does not “experience” its physical system as a curved surface through the 90° translation of the meniscus; it is too thin. The consequence of the connection between the solid metal moving “down” in the z-direction at ~10-3m/s, its overlaying oxide, and the oxide film floating on the molten surface is clear: the oxide on the molten surface is subjected to the same motion in the x-direction as the cast speed in the z-direction. It is also interesting to note that the corners of the rectangular mold cross-section (Figure 14) produce geometrically necessary tears in the oxide and that the deviations away from axisymmetry with the casting speed (e.g. dx/dt = dz/dt = casting speed, dy/dt=0 along the long or rolling face of the mold) could be explained by better coupling of the film across the meniscus near the corners as diagrammed in Figure 14. The oxide near the corners is always undergoing rapid pseudo-plastic behavior because it must be continually torn in the corner of the mold. Oxide near the center of the rolling face is not subject to this geometric consideration. One could argue that machine vibrations, fluid level fluctuations, and other real world process variations play more of a role in how the oxide thickens and when it fractures than in the corner and this leads to a velocity, vRF or (dx/dt)RF near the center of the ingot that is slower than the velocity at the corner, vcorner. This contributes to the characteristic shape of the periodic oxide release seen in Figure 2 and Figure 5. The fluid velocity patterns may also contribute to this shape by providing more organized and linear fluid flow along the same path as shown in Figure 23. The assumptions for gage length of an ideal oxide sheet, its strain rate, and the experimental observations that lend support to these assumptions were discussed at some length in prior sections. The treatment of these topics was cursory at best, but shed some light on the processes involved with the formation of vertical fold defects in Al-Mg4.5 alloys.
(m)
14
Conclusion Vertical folds in Al-Mg4.5 ingots produced by DC casting appear to be a form of wrinkling in thin elastic oxide sheets floating on the aluminum melt. Values for the observed wrinkling frequency λ related to wrinkle length L provide a reasonable – if not exact – correlation to theoretical results when oxidation and pseudo-plasticity effects are allowed. Pseudo-plasticity provides a locally uniform elastic sheet within which wrinkling can occur. Very preliminary calculations show that a transition from pseudo-plastic to brittle behavior is possible over the range of casting speeds and oxide thicknesses observed. As such, pseudo-plasticity is a good candidate cause for the change in oxide behavior from transitory (flowing with the casting speed) to stationary (where vertical folds occur). A number of subtle features of vertical fold formation are supported by wrinkling theory. From a quantitative point of view, the presence of microscopic vertical folds aligned in non-axisymmetric directions is the most robust indicator of wrinkling phenomenon. Fluid flow produces boundary-layer drag forces which cause microscopic wrinkling in the thinnest of oxide films. Formation of vertical folds from skim rings and distribution bags also qualitatively follow from wrinkling theory and are difficult to explain otherwise, although these are not treated here. It is clear, however, that casting speed is the primary driver for wrinkles observed on the macroscopic scale (vertical folds which must be scalped from the ingot). The casting speed imposes a distinct strain rate on the oxide surface due to coupling between the molten head of the ingot and the solid surface below the mold. Once pseudo-plasticity breaks down at a critical oxide thickness, the fracture of the oxide is distinct and brittle and produces large macroscopic wrinkles which we call vertical folds. This process is growth dependent (by the scale laws) and therefore periodic. Given that this is an idealized description of a complex process, we hypothesize that many iterations occur across many different length scales within the oxide mass. This would produce wrinkles on many different lengths since λ ≈ L. Figure 6 shows this behavior as wrinkles are evident from 100 to 10-6 meters in length. Despite the detailed nature of this investigation and the wide functional scope of observations and theoretical treatments, it raises at least as many questions as it answers. To determine precisely the interaction between oxide self-repair and elastic wrinkling theory, more experimental work is required along these or similar lines: a linear or rotational device for oxide films on liquid aluminum should be used to establish ε-dot, to determine the values for l at various strain rates, and to establish the exponent m for pseudo-plasticity. This research will be performed by the authors. Additional theoretical and practical work is encouraged within the mathematical and physical communities for wrinkling physics, boiler tube oxidation, fracture mechanics, thin film growth, and other groups on the periphery of aluminum ingot casting. Oxidation studies under UHV similar to (Stucki, Erbudak et al. 1987) but introducing moist ambient atmospheres and melt temperatures ~700°C would be of great interest to this work.
Acknowledgements The authors would like to thank the following: Novelis Corp. for permission to publish this work; T. Brown at Colorado School of Mines for invaluable laboratory support; Dr. D. Doutre at Novelis Global Technology Center for helpful discussions; T. O’Kelley and his crew at Logan Aluminum for finding the ingot and securing the initial large-scale sectioning; and the casting crews at Novelis Greensboro for producing it.
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