template<int dim, class Scalar = double>
class Kaskade::Elastomechanics::ShiftedInvariants< dim, Scalar >
A class for shifted invariants of a tensor.
For a tensor \( A\in \mathbb{K}^{d\times d} \), the shifted invariants \( i_k \) are just the usual invariants of \( I+A \) shifted by an additive constant. For 2D, they are
\[ i_1 = I_1(I+A)-2 = \mathrm{tr}(I+A)-2 = \mathrm{tr}(A), \quad i_2 = I_2(I+A)-1 = \mathrm{det}(I+A)-1 \]
and for 3D
\[ i_1 = I_1(I+A)-3 = \mathrm{tr}(I+A)-3 = \mathrm{tr}(A), \quad
i_2 = I_2(I+A)-3 = 2\mathrm{tr}(A) + I_2(A) = 2\mathrm{tr}(A) + \frac{1}{2}((\mathrm{tr}(A))^2-\mathrm{tr}(A^2)), \quad
i_3 = I_3(I+A) = \mathrm{det}(I+A)-1. \]
Implementing material laws in terms of shifted invariants of \( 2E \) instead of in the usual invariants of \( C \) has the advantage of numerical stability in the vicinity of the reference configuration.
- Template Parameters
-
| dim | spatial dimension \( d \) |
| Scalar | the type of the underlying field \( K \) |
Definition at line 247 of file elasto.hh.
