Materials¶

Material models describe how the stress in a material depends on its deformation. In a general sense, we can formulate this for an isothermal material abstractly as

\[ \mathbf{P} = \mathcal{F}(\mathbf{F}, \pmb{\alpha}) \]

with a function $\mathcal{F}$ that maps a deformation gradient $\mathbf{F}$ and material state vector $\pmb{\alpha}$ to a stress.

In general, this relation is non-linear and often path-dependent. To solve the global equilibrium equations, the material model must be implemented incrementally. Instead of a total mapping, we evaluate the material response over a discrete load step from time $t_n$ to $t_{n+1}$.

The material model is responsible for advancing the material state. Given the state at the beginning of the step ($\mathbf{F}_n,\mathbf{P}_n, \pmb{\alpha}_n$) and an increment of deformation $\Delta \mathbf{H}$, the model must determine the new stress and updated internal variables ($\mathbf{P}_{n+1}, \pmb{\alpha}_{n+1}$). In addition, it must provide an algorithmic tangent stiffness $$ \mathbb{C}_{n+1} = \frac{\partial \Delta \pmb{\sigma}}{\partial \Delta \mathbf{H}} $$ for convergence speed of the underlying incremental Newton-Raphson solver. In torch-fem, this logic is encapsulated in the step() method of each material. Material holds what every material has -- vectorize(), rotate(), the density, the state width and the spatial dimension -- and one of two bases adds the balance law.

Which material for which model¶

A model takes only a material of its own physics and spatial dimension, and rejects anything else when it is constructed. PS is plane stress and PE plane strain.

Material	Truss	Planar	Solid	Shell	Laminate	TrussHeat	PlanarHeat	SolidHeat	ShellHeat
Isotropic elasticity	1D	PS · PE	3D	PS · PE	PS · PE	—	—	—	—
Orthotropic elasticity	—	PS · PE	3D	PS · PE	PS · PE	—	—	—	—
Transverse isotropic elasticity	—	PS · PE	3D	PS · PE	PS · PE	—	—	—	—
Isotropic plasticity	1D	PS · PE	3D	PS · PE	PS · PE	—	—	—	—
Hyperelasticity	—	PS · PE	3D	—	—	—	—	—	—
Isotropic damage	1D	PS · PE	3D	PS · PE	PS · PE	—	—	—	—
Isotropic conductivity	—	—	—	—	—	1D	2D	3D	2D
Orthotropic conductivity	—	—	—	—	—	—	2D	3D	2D

A hyperelastic material needs geometric nonlinearity, which Shell and Truss do not support, so those cells stay empty whatever material is added.

The step¶

A Mechanics model takes a MechanicsMaterial, whose step maps a displacement gradient increment to a stress:

`torchfem.materials.MechanicsMaterial.step(H_inc, F, stress, state, de0, cl, iter)` `abstractmethod` ¶

Performs an incremental step of the material model.

This function has to update the stress, internal state, and algorithmic tangent stiffness.

Parameters:

H_inc (Tensor) –

Incremental displacement gradient $\Delta \mathbf{H}$. Shape: (..., d, d).
F (Tensor) –

Current deformation gradient $\mathbf{F}_n$. Shape: (..., d, d).
stress (Tensor) –

Current stress tensor $\pmb{\sigma}_n$ or $\mathbf{P}_n$. Shape: (..., d, d).
state (Tensor) –

Internal state variables $\pmb{\alpha}_n$. Shape: (..., <number of state variables>).
de0 (Tensor) –

External strain increment (e.g., thermal). Shape: (..., d, d).
cl (Tensor) –

Characteristic lengths for regularization. Shape: (..., 1).
iter (int) –

Current iteration number.

Returns:

stress_new ( Tensor ) –

Updated stress tensor $\pmb{\sigma}_{n+1}$ or $\mathbf{P}_{n+1}$. Shape: (..., d, d).
state_new ( Tensor ) –

Updated internal state $\pmb{\alpha}_{n+1}$. Shape: (..., n_state).
ddsdde ( Tensor ) –

Algorithmic tangent stiffness tensor $\frac{\partial \Delta \pmb{\sigma}}{\partial \Delta \mathbf{H}}$. Shape: (..., d, d, d, d).

A Heat model takes a HeatMaterial, whose step maps a temperature gradient increment to a heat flux:

`torchfem.materials.HeatMaterial.step(grad_inc, grad, flux, state, cl, iter)` `abstractmethod` ¶