MSAT - University of Bristol

where the repeated indices on the right hand side imply summation in this,
and all similar equations. Similarly, if the stress is known the resulting strain
can be calculated using the compliance tensor which (following the perverse
logic of the stiffness tensor being labelled C) is conventionally denoted S:
ε ij = S ijkl σ kl . (2)
MSAT provides a range of functions for analysing and manipulating representations
of elasticity and these are described below (section 3.2). Many of these
contain implementations of different theories, however we will not reproduce
these here: the reader is directed to the works referenced for that background.
However, there are several pieces of theory which underpin several of the functions
provided, which are worth covering briefly.
The first of these is the symmetry exhibited by the elastic constants tensor.
The symmetry of the stress and strain tensors implies that C ijkl = C jikl and
C ijkl = C ijlk (the minor symmetries). Furthermore, it turns out that the
stiffness tensor can be written as a function of the second derivative of the
internal energy, U, with strain:
C ijkl =
∂2 U
∂ε ij ∂ε kl
, (3)
and this results in the major symmetries: C ijkl = C klij . Taken together these
symmetry relations reduce the 81 constants relating 9 stresses to 9 strains to
21 distinct constants relating 6 stresses to 6 strains and these arguments apply
equally to S. For convenience, elastic constants are usually representation using
Voigt notation. This represents the 4th order tensor C ijkl as a symmetric 6×6
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