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Reference documentation for deal.II version 8.5.1
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Namespaces | |
Kinematics | |
Classes | |
class | StandardTensors |
This namespace provides a collection of definitions that conform to standard notation used in (nonlinear) elasticity.
References for this notation include:
For convenience we will predefine some commonly referenced tensors and operations. Considering the position vector in the referential (material) configuration, points
are transformed to points
in the current (spatial) configuration through the nonlinear map
where the represents the displacement vector. From this we can compute the deformation gradient tensor as
wherein the differential operator is defined as
and
is the identity tensor.
Finally, two common tensor operators are represented by and
operators. These respectively represent a single and double contraction over the inner tensor indices. Vectors and second-order tensors are highlighted by bold font, while fourth-order tensors are denoted by calliagraphic font.
One can think of fourth-order tensors as linear operators mapping second-order tensors (matrices) onto themselves in much the same way as matrices map vectors onto vectors. To provide some context to the implemented class members and functions, consider the following fundamental operations performed on tensors with special properties:
If we represent a general second-order tensor as , then the general fourth-order unit tensors
and
are defined by
or, in indicial notation,
with the Kronecker deltas taking their common definition. Note that .
We then define the symmetric and skew-symmetric fourth-order unit tensors by
such that
The fourth-order symmetric tensor returned by identity_tensor() is .