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Orthotropic Elasticity Plane Strain

Bases: OrthotropicElasticity3D

Orthotropic elastic material for plane strain problems.

Parameters:

  • E_1 (Tensor | float) –

    Young's modulus in direction 1.

  • E_2 (Tensor | float) –

    Young's modulus in direction 2.

  • E_3 (Tensor | float) –

    Young's modulus in direction 3.

  • nu_12 (Tensor | float) –

    Poisson's ratio \(\nu_{12}\).

  • nu_13 (Tensor | float) –

    Poisson's ratio \(\nu_{13}\).

  • nu_23 (Tensor | float) –

    Poisson's ratio \(\nu_{23}\).

  • G_12 (Tensor | float) –

    In-plane shear modulus \(G_{12}\).

  • G_13 (Tensor | float, default: 0.0 ) –

    Transverse shear modulus \(G_{13}\). Default is 0.0.

  • G_23 (Tensor | float, default: 0.0 ) –

    Transverse shear modulus \(G_{23}\). Default is 0.0.

  • rho (Tensor | float, default: 1.0 ) –

    Mass density. Default is 1.0.

Notes
  • Small-strain assumption with plane strain condition (\(\varepsilon_{33} = 0\)).
  • No internal state variables (n_state = 0).
  • Supports rotation of the material coordinate system via rotate().
Plane strain orthotropic stiffness

The in-plane stiffness tensor is derived from the full 3D orthotropic stiffness by enforcing \(\varepsilon_{33} = 0\). The in-plane components are populated using the same reciprocal relations as the 3D case, with the full factor \(F = (1 - \nu_{12}\nu_{21} - \nu_{13}\nu_{31} - \nu_{23}\nu_{32} - 2\nu_{21}\nu_{32}\nu_{13})^{-1}\).

vectorize(n_elem)

Returns a vectorized copy of the material for n_elem elements.

Parameters:

  • n_elem (int) –

    Number of elements to vectorize the material for.

Returns:

rotate(R)

Rotates the material coordinate system with a rotation matrix \(\mathbf{R}\).

Parameters:

  • R (Tensor) –

    Rotation matrix. Shape: (..., 2, 2).

Returns:

step(H_inc, F, stress, state, de0, cl, iter)

Performs an incremental step in the orthotropic elasticity model.

The stress is updated as $$ \pmb{\sigma}_{n+1} = \pmb{\sigma}_n + \mathbb{C} : (\Delta \pmb{\varepsilon} - \Delta \pmb{\varepsilon}^0) $$

Parameters:

  • H_inc (Tensor) –

    Incremental displacement gradient. Shape: (..., 3, 3).

  • F (Tensor) –

    Current deformation gradient. Shape: (..., 3, 3).

  • stress (Tensor) –

    Current Cauchy stress tensor. Shape: (..., 3, 3).

  • state (Tensor) –

    Internal state variables (unused). Shape: (..., 0).

  • de0 (Tensor) –

    External small strain increment (e.g., thermal). Shape: (..., 3, 3).

  • cl (Tensor) –

    Characteristic lengths. Shape: (..., 1).

  • iter (int) –

    Current iteration number.

Returns:

  • stress_new ( Tensor ) –

    Updated Cauchy stress tensor. Shape: (..., 3, 3).

  • state_new ( Tensor ) –

    Updated internal state (unchanged). Shape: (..., 0).

  • ddsdde ( Tensor ) –

    Algorithmic tangent stiffness tensor. Shape: (..., 3, 3, 3, 3).