Viscoplastic constitutive equations
Anisotropic power-law rheologies are supported in both forward and inverse (or reverse) form.
Eigenenhancements
Bulk viscous anisotropy is prescribed in terms of logitudinal and shear strain-rate enhancement factors w.r.t rheological symmetry axes, termed eigenenhancements (\(E_{ij}\)).
Transversely isotropic
| Rheolgical symmetries | Forward rheology |
|---|---|
 |
$$ {\bf D} = \eta^{-1} \Big( {\bf S} - \lambda_1 ({\bf S}:{\bf M}){\bf I} + \lambda_2 ({\bf S}:{\bf M}){\bf M} + \lambda_3 ({\bf S}\cdot{\bf M} + {\bf M}\cdot{\bf S}) \Big) $$ $$ \eta^{-1} = A\Big( {\bf S}:{\bf S} + \lambda_2 ({\bf S}:{\bf M})^2 + 2\lambda_2 ({\bf S}^2:{\bf M}) \Big)^{(n-1)/2} $$ $$ {\bf M}={\bf m}^2$$ |
where the material parameters \(\lambda_i\) depend on the eigenenhancements:
\[
\lambda_1 = \frac{E_{mm}^{2/(n+1)}-1}{2} ,\quad
\lambda_2 = \frac{3(E_{mm}^{2/(n+1)}-1) - 4(E_{mt}^{2/(n+1)}-1)}{2} ,\quad
\lambda_3 = E_{mt}^{2/(n+1)}-1
\]
API
D = sf.rheo_fwd_tranisotropic(S, A, n, m, Eij)
S = sf.rheo_rev_tranisotropic(D, A, n, m, Eij)
where
| Arguments | Type |
|---|---|
S, D |
Deviatoric-stress and strain-rate tensor (3x3) |
A, n |
Flow-rate factor \(A\) and power-law exponent \(n\) |
m |
Rotational symmetry axis \(\bf{m}\) |
Eij |
Tuple of eigenenhancements (Emm, Emt) |
Orthotropic
| Rheolgical symmetries | Forward rheology |
|---|---|
 |
\({\bf D} = \eta^{-1} \displaystyle\sum_{i=1}^{3} \Big[ \lambda_i ({\bf S}:{\bf M}_i){\bf M}_{i} + \lambda_{i+3} ({\bf S}:{\bf M}_{i+3}) {\bf M}_{i+3} \Big]\) \(\eta^{-1} = A\left( \displaystyle\sum_{i=1}^3 \Big[ \lambda_i ({\bf S}:{\bf M}_{i})^2 + \lambda_{i+3} ({\bf S}:{\bf M}_{i+3})^2 \Big] \right)^{(n-1)/2}\) \({\bf M}_{i} = \dfrac{{\bf m}_{j_i} {\bf m}_{j_i} - {\bf m}_{k_i} {\bf m}_{k_i}}{2} ,\quad {\bf M}_{i+3} = \dfrac{{\bf m}_{j_i} {\bf m}_{k_i} + {\bf m}_{k_i} {\bf m}_{j_i}}{2},\) \((j_1, j_2, j_3) = (2,3,1),\quad (k_1, k_2, k_3) = (3,1,2)\) |
where the material parameters \(\lambda_i\) depend on the eigenenhancements:
\[
\lambda_i = \frac{4}{3} \left( E_{j_i j_i}^{2/(n+1)} + E_{k_i k_i}^{2/(n+1)} - E_{i i}^{2/(n+1)} \right),\quad
\lambda_{i+3} = 2 E_{j_i k_i}^{2/(n+1)}
\]
API
D = sf.rheo_fwd_orthotropic(S, A, n, m1,m2,m3, Eij)
S = sf.rheo_rev_orthotropic(D, A, n, m1,m2,m3, Eij)
where
| Arguments | Type |
|---|---|
S, D |
Deviatoric-stress and strain-rate tensor (3x3) |
A, n |
Flow-rate factor \(A\) and power-law exponent \(n\) |
m1, m2, m3 |
Reflection symmetry axes \(\bf{m}_1\), \(\bf{m}_2\), and \(\bf{m}_3\) |
Eij |
Tuple of eigenenhancements (E11,E22,E33,E23,E13,E12) |