19-01-2013, 12:35 PM
Matrix and tensor notation in the theory of elasticity
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Abstract
This summary1 focuses on the connection between the tensor and the matrix notation
in the theory of elasticity. Many theories in this field are based on the manipulation of
tensorial equations, e.g. the rotation of the stiffness or the compliance tensor, or calculating
the inverse of fourth order symmetric tensors. The focus of the second part of this summary
is the application of the matrix notation to the theories of porous materials.
The Basics
The theory of elastic constants is a theory of two second–order symmetric tensors (strain " and
stress ¾) and four fourth–order symmetric tensors (the stiffness tensor C, the compliance tensor
D, the Eshelby tensor S and its counterpart T.
Second–order symmetric tensors and fourth–order symmetric tensors can be constructed from
second–order symmetric base tensors es
Rotations of the coordinate system
Sometimes it is necessary to calculate the elastic constants or strain and shear in a rotated
coordinate system. The transformation
S0
kl = SijLikLjl (41)
converts the symmetric second–order tensor Slm into its representation in the new coordinate
system S0
ik. The Lij are orthogonal symmetric second–order tensors, whose components are the
directions cosines between the i-th axis of the new and the j-th axis of the old coordinate axes.
The direction cosines are equal to the scalar product of a unit vector in the direction of the i-th
axis of the old system with a unit vector in the direction of the j-th axis of the new system. If
~xi are the unit vectors of the old system and ~yj are the unit vectors of the new system, then
Lij = ~xi · ~yj .
The matrix representation of the second–order tensors Lij cannot be easily used to transform
the matrix notation of Slm. What is needed is a symmetric fourth–dimensional tensor, whose
double contraction is equivalent to the two single contraction with the tensors Lik and Ljl. We
therefore define the symmetric fourth–order tensor Aijkl.