3.1.6.19.3. Elasticity Types

These components define the stress-strain relationship for the linear part of the model (see ASDPlasticMaterial Theory).

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

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

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

Parameter name	Type	Symbol	Description
YoungsModulus	scalar	\(E\)	Young’s modulus
PoissonsRatio	scalar	\(\nu\)	Poisson’s ratio

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
ReferencePressure	scalar	\(p_{ref}\)	Reference pressure for the definition of Young’s modulus.
DuncanChang_MaxSigma3	scalar	\(\sigma_{3max}\)	Maximum confinement stress.
DuncanChang_n	scalar	\(\nu\)	Exponent for Duncan-Chang law.