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Isotropic Plasticity 3D

Bases: IsotropicElasticity3D

Isotropic elastoplastic material model in 3D.

This class extends IsotropicElasticity3D to incorporate isotropic plasticity with a von Mises yield criterion and associative flow rule.

Parameters:

  • E (Tensor | float) –

    Young's modulus. Shape: () for a scalar or (N,) for a batch of materials.

  • nu (Tensor | float) –

    Poisson's ratio. Shape: () for a scalar or (N,) for a batch of materials.

  • sigma_f (Callable) –

    Yield stress function \(\sigma_f(q)\) of the equivalent plastic strain \(q\).

  • sigma_f_prime (Callable) –

    Derivative \(\sigma_f'(q)\).

  • tolerance (float, default: 1e-05 ) –

    Convergence tolerance for the local Newton solver. Default is 1e-5.

  • max_iter (int, default: 10 ) –

    Maximum number of Newton iterations. Default is 10.

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

    Mass density. Default is 1.0.

Notes
  • Small-strain assumption.
  • One internal state variable (n_state = 1): equivalent plastic strain \(q\).
  • Supports batched/vectorized material parameters.
Von Mises plasticity

The elastic domain is bounded by the yield surface $$ f(\pmb{\sigma}, q) = |\pmb{\sigma}'| - \sqrt{\tfrac{2}{3}} \, \sigma_f(q) = 0 $$ with the deviatoric stress \(\pmb{\sigma}'\) and the yield stress \(\sigma_f(q)\), given as a function of the equivalent plastic strain \(q\). Linear hardening is \(\sigma_f(q) = \sigma_y + k q\) with the initial yield stress \(\sigma_y\) and the hardening modulus \(k\).

A state with \(f > 0\) is returned to the surface along the associative flow direction \(\mathbf{n} = \pmb{\sigma}' / \|\pmb{\sigma}'\|\), which is the radial return step(...) carries out.

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.

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

Performs a strain increment with the radial return-mapping algorithm.

In each increment, a trial stress is computed as

\[ \pmb{\sigma}_{\text{trial}} = \pmb{\sigma}_n + \mathbb{C} : \Delta\pmb{\varepsilon} \]

and the yield condition is checked via the flow potential

\[ f = \|\pmb{\sigma}'_{\text{trial}}\| - \sqrt{\tfrac{2}{3}} \, \sigma_f(q) \]

where \(\pmb{\sigma}'\) denotes the deviatoric stress.

Elastic step (\(f \le 0\)): The trial stress is accepted.

Plastic step (\(f > 0\)): The plastic multiplier \(\Delta\gamma\) that restores \(f = 0\) solves a nonlinear equation, found by a local Newton iteration bounded by tolerance and max_iter, and reached in one step for linear hardening. The flow direction is \(\mathbf{n} = \pmb{\sigma}'_{\text{trial}} / \|\pmb{\sigma}'_{\text{trial}}\|\) and the updates are

\[ \pmb{\sigma}_{n+1} = \pmb{\sigma}_{\text{trial}} - 2 G \, \Delta\gamma \, \mathbf{n}, \quad q_{n+1} = q_n + \sqrt{\tfrac{2}{3}} \, \Delta\gamma. \]

The consistent (algorithmic) tangent is

\[ \mathbb{C}^{\text{alg}} = \mathbb{C} - \frac{2 G}{1 + \frac{\sigma_f'}{3 G}} \, \mathbf{n} \otimes \mathbf{n} - \frac{4 G^2 \Delta\gamma} {\|\pmb{\sigma}'_{\text{trial}}\|} \left( \mathbb{I}^{\text{dev}} - \mathbf{n} \otimes \mathbf{n} \right) \]

with \(\mathbb{I}^{\text{dev}}_{ijkl} = \frac{1}{2}( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) - \frac{1}{3}\delta_{ij}\delta_{kl}\).

Parameters:

  • H_inc (Tensor) –

    Incremental displacement gradient. Shape: (..., 3, 3), where ... represents batch dimensions.

  • F (Tensor) –

    Current deformation gradient. Shape: (..., 3, 3), same as H_inc.

  • stress (Tensor) –

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

  • state (Tensor) –

    Internal state variables, here: equivalent plastic strain. Shape: (..., 1).

  • 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 after plastic update. Shape: (..., 3, 3).

  • state_new ( Tensor ) –

    Updated internal state with updated plastic strain. Shape: same as state.

  • ddsdde ( Tensor ) –

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