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Magic
0.8
What do cement floors, earthquakes
and broken cups have in common? A magic number.
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Alex Hansen
is looking for magic numbers.
Contact: Alex Hansen, Department of Physics, NTNU
Tel: +47 73 59 36 49
Email:alex.hansen@phys.ntnu.no
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| Photo: Rune Petter Ness |
In 1975, the Polish-born mathematician Benoit
Mandelbrot made a sensational discovery: when he broke aluminium
poles of different thicknesses in half and examined the fracture
surfaces on the poles, he discovered to his astonishment that the
same fracture patterns appeared again and again, regardless of the
fracture surface size. At the beginning of the 1990s, two groups
of physicists independently published new research results, advancing
Mandelbrot’s observations.
Their results not only showed that the same
patterns appeared no matter the size, but that they were governed
by one specific, ‘magic’ number or mathematical constant
that the researchers claimed applied to all brittle materials. Whether
a brick cut in half, or a mountain crack, the fracture pattern could
be described with the same number: 0.8. The researchers claimed
that if they used this number in a certain formula, they could calculate
the behaviour of cracks in all brittle materials, such as brick
walls, cast-iron and porcelain.
Their claim was based both on experiments
and on numerical calculations made with computer models. In spite
of the empirical data, no one could offer a deep, theoretical understanding
of this discovery. In subsequent years, many scientists tried to
explain the theory behind this magic number, but up until recently,
nobody had been able to crack the code.
CALCULATIONS
Professor Alex Hansen, now a physicist in NTNU’s Physics Department,
participated in this pioneering work in the 1990s. Today the professor
and two French research colleagues claim they have finally found
the answer as to how this ‘magic’ number came into being.
For the first time, researchers have managed to calculate the number
they have previously seen only in experiments.
Researchers at the University of Oslo have
conducted the physical experiments that support the theory. Some
pieces are still missing from the puzzle, but the most important
evidence is in Professor Hansen’s hands. The results have
gained international attention and have been published in the renowned
journal Physical Review Letters.
FRACTURE HERESY
Scepticism about the new idea is still widespread among those who
have traditionally worked with fracture surfaces – mechanical
engineers. Prof. Hansen says that the claim that certain characteristics
remain the same, regardless of the material, is heresy in traditional
fracture research. Nevertheless, the theory has gradually been accepted
by researchers all over the world. In fact, Prof. Hansen has been
asked to arrange a symposium about the findings at a large, international
conference on fractures in Italy in 2005, arranged by mechanical
engineers.
Professor Hansen says the most important
aspect of these discoveries is that they allow for the transfer
of results from laboratories to full-scale systems. For example,
how do you transfer the results from metal plates measuring 20x20
cm to plates measuring 20x20 metres? This problem of upscaling is
very old and, according to Prof. Hansen, it will continue to be
a major and important challenge.
PREDICTS SHAPE
What happens when a material is exposed to external stress? When
will it crack? Where will the crack develop and where will it end?
What will it look like? In order to conduct the numerical experiments
needed to answer these questions, Prof. Hansen has employed what
physicists call a fuse model. A network of fuses – the same
fuses found in your fuse box – is arranged much like an advanced
computer program.
The fuses can take varying amperage, and
as the current is increased, the smallest fuses blow, forcing the
current to find its way around the broken fuses. That increases
the pressure on the other fuses, and they blow as well. Finally,
the entire network breaks down. The fuses represent strong and weak
areas in a material and make it possible to predict the shape of
the cracks. In brittle materials, the process happens like this:
Numerous micro cracks develop gradually in the material.
These micro cracks will change the distribution
of forces in the material, and these forces cause new micro cracks.
Hence, there is a circular connection between the micro cracks and
the forces, and eventually, the material cracks. This circular connection
takes complete control and overrides specific fracture characteristics
of different materials. Thus, the characteristics of each material
become irrelevant, making the same magic number applicable to all
brittle materials.
HELP FOR EARTHQUAKE RESEARCHERS
The discovery makes it possible to transfer small-scale laboratory
tests to large-scale reallife situations. For example, the theory
can give seismologists a better understanding of how faults (cracks
in the Earth’s crust) behave during earthquakes, and therefore
can contribute to a more accurate warning of seismic dangers. In
September 2004, Kobe, Japan was struck by two earthquakes.
The first quake was measured at 8.0 on the
Richter scale and was the most powerful earthquake in Japan for
almost nine years. Measurements conducted after the quake showed
that the faults behaved according to the theory developed by Hansen
and his French colleagues. Earthquakes mainly occur on the border
between the plates of the Earth’s crust. Tensions between
these plates are highly affected by their roughness (the shape of
their fracture surfaces) and by the stress peaks they create.
“Our research makes it easier to understand
this roughness, and beyond that, the processes triggering earthquakes.
By measuring the tensions in a given area,we can predict the tension
other places along the fault. Knowing this, we will be able to predict
where an earthquake of a given strength will occur. This could be
invaluable knowledge to future seismologists,” concludes Hansen.
Elin Fugelsnes
SAME MODEL
FOR POWER FAILURE
The physicists’ theory not only applies to fractures.
It also applies to power failures.
In August 2003, an enormous swath of eastern North America,
extending from New York City northeast to Toronto, Canada,
was struck by an extensive and prolonged power failure that
ultimately affected more than 50 million people. A month later,
all of Italy was blacked out. How could 55 million Italians
lose their power simultaneously?
Jan Øystein Bakke, NTNU research fellow at the Department
of Physics, has collected data from several power US failures
in the US. He hopes they can reveal some of the mystery. The
data describes the extent of the different failures and the
electric grid structure.
One of the tools Mr. Bakke uses to make theoretical predictions
about when the grid will break down is the same fuse model
Professor Hansen uses in his simulations. However, Mr. Bakke’s
doesn’t start his analysis with fuse networks. Instead,
he uses actual, irregular networks from Scandinavia, where
the flow is unevenly distributed over the network. When Mr.
Bakke studied how power failures travel from one minor failure
to another, and then lead to a major failure, it turns out
that the rules are the same as for earthquakes.
A comparison of the number of failures or earthquakes of
a given strength with failures or earthquakes of differing
strengths, shows that that the proportions are statistically
identical. “Earthquakes and power failure are rare events.
Our research is trying to produce a systematic way to calculate
the statistical probability of these events,” Hansen
says. |
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