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Transverse Isotropic Elasticity Plane Strain

Bases: OrthotropicElasticityPlaneStrain

Transversely isotropic elastic material for plane strain problems.

A special case of OrthotropicElasticityPlaneStrain where the in-plane direction 1 is the axis of symmetry and the transverse plane (2-3) is isotropic.

Parameters:

  • E_L (Tensor | float) –

    Longitudinal Young's modulus.

  • E_T (Tensor | float) –

    Transverse Young's modulus.

  • nu_L (Tensor | float) –

    Longitudinal Poisson's ratio \(\nu_L = \nu_{12} = \nu_{13}\).

  • nu_T (Tensor | float) –

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

  • G_L (Tensor | float) –

    Longitudinal shear modulus \(G_L = G_{12} = G_{13}\).

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

    Mass density. Default is 1.0.

Notes
  • Only five independent constants: \(E_L, E_T, \nu_L, \nu_T, G_L\).
  • The transverse shear modulus is derived as \(G_T = E_T / (2(1 + \nu_T))\).
  • Raises ValueError if \(G_L > E_L / (2(1 + \nu_L))\).
Transverse isotropy as a special case

Isotropy in the transverse plane (2-3) reduces the nine independent constants of OrthotropicElasticityPlaneStrain to five, $$ E_1 = E_L, \quad E_2 = E_3 = E_T $$ $$ \nu_{12} = \nu_{13} = \nu_L, \quad \nu_{23} = \nu_T $$ $$ G_{12} = G_{13} = G_L, \quad G_{23} = G_T = \frac{E_T}{2 (1 + \nu_T)} $$ where the transverse shear modulus \(G_T\) follows from isotropy in the 2-3 plane. The plane strain condition \(\varepsilon_{33} = 0\) is then enforced by OrthotropicElasticityPlaneStrain.

vectorize(n_elem)

Returns the material batched over n_elem elements.

Parameters:

  • n_elem (int) –

    Number of elements to vectorize the material for.

Returns:

  • Material ( T ) –

    A material of the same type carrying one entry per element, or itself if it is vectorized already.

rotate(R)

Returns a copy with its coordinate system rotated by \(\mathbf{R}\).

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).