Elasticity Types

These components define the stress-strain relationship for the linear part of the model (see ASDPlasticTheory). .. math:

\newcommand{\vec}[1]{\boldsymbol{#1}}
\vec{\sigma} = \vec{E} \vec{\epsilon}

Where the operator \(\vec{E}\) may depend on the material state, parameters, etc.

Note

When using ASDPlasticMaterial3D, elastic parameters can be updated during analysis using the setParameter command, allowing for simulation of materials with time-dependent or state-dependent elastic properties. This is particularly useful for modeling damage effects, temperature-dependent stiffness, or staged construction processes.

LinearIsotropic3D_EL

A classic!

\[\begin{split}\vec{E}\,=\,{\frac {E}{(1+\nu )(1-2\nu )}}{\begin{bmatrix}1-\nu &\nu &\nu &0&0&0\\\nu &1-\nu &\nu &0&0&0\\\nu &\nu &1-\nu &0&0&0\\0&0&0&{\frac {1-2\nu }{2}}&0&0\\0&0&0&0&{\frac {1-2\nu }{2}}&0\\0&0&0&0&0&{\frac {1-2\nu }{2}}\end{bmatrix}}\end{split}\]

Parameters required

Usage in ASDPlasticMaterial3D

LinearIsotropic3D_EL provides the fundamental elastic behavior for most engineering materials. With ASDPlasticMaterial3D:

  • Constant elastic properties throughout analysis

  • Efficient computation with closed-form tangent

  • Suitable for metals, concrete at small strains, and other linear elastic materials

Monitor elastic behavior:

# Update elastic parameters during analysis
setParameter 1 YoungsModulus 3e10
setParameter 1 PoissonsRatio 0.2

DuncanChang_EL

This is a hypoelastic model that features pressure dependent behavior. It is composed of an isotropic elastic model where the Young’s modulus has the following dependency on the maximum principal stress \(\sigma_3\) (it is assumed that \(\sigma_1 \leq \sigma_2 \leq \sigma_3 < 0\)).

\[E(\sigma_3) = E_{ref} \cdot p_{ref} \cdot \left(\dfrac{\vert \sigma_3 \vert}{ p_{ref}} \right)^n\]

Where \(E_{ref}\) (dimensionless) specifies the Young’s modulus at reference pressure \(p_{ref}\) and \(n\) is a material constant. A cut-off maximum confinement pressure \(\sigma_{3max}\) at which the Young’s modulus will be evaluated should the confinement be greater than that value.

Parameters required

Parameter name

Type

Symbol

Description

ReferenceYoungsModulus

scalar

\(E_{ref}\)

Dimensionless reference Young’s modulus at reference pressure \(p_{ref}\).

PoissonsRatio

scalar

\(\nu\)

Poisson’s ratio (dimensionless)

ReferencePressure

scalar

\(p_{ref}\)

Reference pressure for the definition of Young’s modulus (Pa).

DuncanChang_MaxSigma3

scalar

\(\sigma_{3max}\)

Maximum confinement stress (Pa).

DuncanChang_n

scalar

\(n\)

Exponent for Duncan-Chang law (typically 0.2-1.0).

Usage in ASDPlasticMaterial3D

DuncanChang_EL provides pressure-dependent elastic behavior, particularly useful for:

  • Soil materials with stiffness increasing with confinement

  • Rock materials exhibiting non-linear elastic response

  • Interfaces and joints with pressure-dependent behavior

The Young’s modulus varies with minimum principal stress \(\sigma_3\) (compression negative):

\[E(\sigma_3) = E_{ref} \cdot p_{ref} \cdot \left(\dfrac{\vert \sigma_3 \vert}{ p_{ref}} \right)^n\]

Monitor pressure-dependent response:

recorder Node -file pressure.out -time -node 1 -dof 1 PStress
recorder Node -file j2_stress.out -time -node 1 -dof 1 J2Stress