Namespaces | |
namespace | 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
where the
wherein the differential operator
Finally, two common tensor operators are represented by
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
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