Solid Mechanics Model¶

Introduction¶

The solid-mechanics model describes the mechanical response of solid bodies under applied loads in the small-strain, quasi-static regime. It solves for the displacement field $\mathbf{u}(\mathbf{x})$ in static equilibrium

\[ \nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \mathbf{0} \qquad \text{in } \Omega, \]

with the infinitesimal strain tensor

\[ \boldsymbol{\varepsilon}(\mathbf{u}) = \tfrac{1}{2} \left( \nabla \mathbf{u} + \nabla \mathbf{u}^{\mathsf{T}} \right) \]

and the isotropic linear-elastic stress–strain relation

\[ \boldsymbol{\sigma} = 2 \mu \, \boldsymbol{\varepsilon} + \lambda \, \mathrm{tr}(\boldsymbol{\varepsilon}) \, \mathbf{I}, \]

where the Lamé parameters are derived from Young's modulus $E$ and Poisson's ratio $\nu$:

\[ \mu = \frac{E}{2(1+\nu)}, \qquad \lambda = \frac{E \, \nu}{(1+\nu)(1-2\nu)}. \]

The model has typical industrial applications:

Cantilever and pressure-vessel analysis.
Support-bracket sizing.
Mechanical-strength verification of electronics enclosures.
The structural arm of magnet–mechanical coupling (motor housings, MRI gradient coils, magnet brackets).

Model¶

You create the model and add it to the simulation with

structural_model = StructuralModel(
    marker=structural_domain,
    order=2,
)

sim.get_model_manager().add_model(structural_model)

where

marker specifies the domain on which the model is solved (defaults to the whole volume),
order is the polynomial order of the displacement discretisation (default 1). $p = 2$ is a common choice for bending-dominated problems where linear elements suffer from shear locking.

The model has these methods to manage its setup: add_material, add_materials, add_condition, add_conditions, get_materials, get_conditions, clear_materials, clear_conditions, and get_marker. The clear_* methods remove all materials or conditions from the model.

Materials¶

The model currently supports a single constitutive law — the Linear Elastic Material — characterised by Young's modulus $E$ and Poisson's ratio $\nu$:

from mufem.structural import LinearElasticMaterial

steel = LinearElasticMaterial(
    name="Steel",
    marker="Beam" @ Vol,
    youngs_modulus=210.0e9,
    poissons_ratio=0.30,
)

structural_model.add_material(steel)

Conditions¶

List of supported conditions
Name	Supported Entities	Description
Fixed Displacement	Boundary	Clamps the displacement to zero on the boundary (homogeneous Dirichlet, $\mathbf{u} = \mathbf{0}$).
Traction	Boundary	Prescribes a surface traction $\boldsymbol{\sigma}\,\mathbf{n} = \bar{\mathbf{t}}$ (inhomogeneous Neumann).

Coefficients¶

List of functions
Name	Field Type	Description
Displacement	Vector	The displacement field $\mathbf{u}$ [m].
Von Mises Stress	Scalar	The von Mises equivalent stress $$ \sigma_\mathrm{vM} = \sqrt{\tfrac{3}{2}\, \boldsymbol{s} : \boldsymbol{s}}, $$ where $\boldsymbol{s} = \boldsymbol{\sigma} - \tfrac{1}{3}\mathrm{tr}(\boldsymbol{\sigma})\,\mathbf{I}$ is the deviatoric stress. Standard yield criterion for ductile metals.