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Mechanics of Magnetic Fusion Reactors Chapter | 12 367

where r is the radius vector from an arbitrary point in a current configuration
space of volume V; µ is the absolute magnetic permeability; and r is the radius
1
0
vector of elementary area dV of the charged conductor surface.
In most cases, it is sufficient to calculate relevant forces using a linear
(undeformed) model. Additional magnetic forces due to the deformation of
current-carrying elements are usually treated as ‘minor additions’. However,
the evolution of effective forces associated with the current-carrying elements’
deviation from their initial direction is an aspect of key importance for the sta-
bility analysis. In addition, MS elements may be subject to displacements that
give rise to forces that were absent in the initial configuration and can hardly be
regarded as ‘minor additions’ to forces already in place. In this case, changes in
magnetic fields and forces occurring during the deformation process should be
accounted for to ensure adequate stability assessment.
An increased attention to the problem of magneto-elastic stability was
largely the result of the development of large fusion magnets [9]. The problem
became particularly relevant with the transition to superconducting coils and
application of strengthening structural steels. This allowed higher mechanical
stresses to be applied to load-bearing structures and a greater elastic energy to
be stored in an MS. Potential loss of a TFC magneto-elastic stability was pre-
dicted theoretically and demonstrated on mockups [10].
Where there is a minor deformation of current-carrying elements, additional
electromagnetic forces are proportional to the displacement, and the equilib-
rium matrix equation takes the form
)
c ( e + c u = F , (ce+cm)u=F ,
m
0
0
where c is the linear elasticity, c is the magnetic stiffness (see Appendix
e
m
A.12.1), and F is for the ponderomotive forces acting on the original (unde-
0
formed) structure. Where there is a deviation from this state, additional forces
due to magnetic stiffness either revert the MS back to its original state (the
positive magnetic stiffness case) or, vice versa, increase the deformation, mov-
ing the MS away from its original state (the negative magnetic stiffness case),
depending on the MS configuration and the current direction. Meanwhile, Earn-
shaw’s theorem implies that in a system with direct currents, displacements as-
sociated with negative magnetic stiffness always occur. Thus, an MS inevitably
becomes unstable if elastic restoring forces are smaller than the destabilising
ponderomotive forces. Potentiality of the forces allows the Euler stability crite-
rion to be applied to our analysis. In this context, a system with many degrees
of freedom is uniquely stable, if a matrix composed of elastic and magnetic
stiffness constituents is positively definite.

12.5.2 Stability of Toroidal and Poloidal Field Systems
Let us treat the above issues in more detail using a present-day tokamak as an
example. A system of D-shaped toroidal field coils makes up a compact torus.