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Linear Elastic Material

LinearElasticMaterial defines an isotropic linear-elastic solid on a marked region. It is characterised by two scalar parameters:

  • \(E\) — Young's modulus [Pa]
  • \(\nu\) — Poisson's ratio [dimensionless]

The stress–strain relation is

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

with the Lamé parameters

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

Typical applications

  • Structural metals — steel (\(E \approx 210\,\mathrm{GPa}\), \(\nu \approx 0.30\)), aluminium (\(E \approx 70\,\mathrm{GPa}\), \(\nu \approx 0.33\)), titanium, copper.
  • Engineering polymers within their small-strain regime (PMMA, ABS, epoxy laminates).
  • Concrete and masonry for first-order load studies away from cracking.
  • Ceramics (alumina, silicon carbide) where brittle failure rather than yielding governs and von-Mises is used as a strength proxy.

The model is small-strain and linear-elastic — it is not appropriate for problems with large deformations, plastic yielding, or material non-linearity.

Usage

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)

Multiple materials can be assigned to disjoint sub-regions of the same structural model — for example a steel frame with an aluminium plate.