tensors

Basic tensors

class mechkit.tensors.Basic

Basic tensors in tensor notations

I2

Identity on second order tensors

\[\begin{align*} \mathbf{I} &= \delta_{ij} \; \mathbf{e}_{i} \otimes \mathbf{e}_{j} \end{align*}\]

Type:

np.array of shape (3, 3,)

I4

Identity on fourth order tensors

\[\begin{split}\begin{align*} \mathbb{I} &= \delta_{ik} \delta_{lj} \; \mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes \mathbf{e}_{k} \otimes \mathbf{e}_{l} \\ \end{align*}\end{split}\]

Type:

np.array of shape (3, 3, 3, 3,)

I4s

Identity on symmetric fourth order tensors

\[\begin{split}\begin{align*} \mathbb{I}^{\text{S}} &= \frac{1}{2} \left( \mathbb{I} + \mathbb{I}^{\text{T}_\text{R}} \right) \\ &= \frac{1}{2} \left( \delta_{ik} \delta_{lj} + \delta_{il} \delta_{kj} \right) \; \mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes \mathbf{e}_{k} \otimes \mathbf{e}_{l} \\ \end{align*}\end{split}\]

Type:

np.array of shape (3, 3, 3, 3,)

I4a

Identity on asymmetric fourth order tensors

\[\begin{split}\begin{align*} \mathbb{I}^{\text{A}} &= \frac{1}{2} \left( \mathbb{I} - \mathbb{I}^{\text{T}_\text{R}} \right) \\ &= \frac{1}{2} \left( \delta_{ik} \delta_{lj} - \delta_{il} \delta_{kj} \right) \; \mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes \mathbf{e}_{k} \otimes \mathbf{e}_{l} \\ \end{align*}\end{split}\]

Type:

np.array of shape (3, 3, 3, 3,)

P1

First isotropic projector. Projecting second order tensor onto its spherical part

\[\begin{split}\begin{align*} \mathbb{P}_{\text{1}} &= \frac{1}{3} \mathbf{I} \otimes \mathbf{I} \\ &= \frac{1}{3} \delta_{ij} \delta_{kl} \; \mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes \mathbf{e}_{k} \otimes \mathbf{e}_{l} \end{align*}\end{split}\]

Type:

np.array of shape (3, 3, 3, 3,)

P2

Second isotropic projector. Projecting second order tensor onto its symmetric deviatoric part

\[\begin{align*} \mathbb{P}_{\text{2}} &= \mathbb{I}^{\text{S}} - \mathbb{P}_{\text{1}} \end{align*}\]

Type:

np.array of shape (3, 3, 3, 3,)