Model::Solid::Linear::Isotropic
===============================

:bdg-link-secondary-line:`src/Model/Solid/Linear/Isotropic.H <../../../../doxygen/Linear_2Isotropic_8H_source.html>`

This model implements an isotropic linear elastic material.
See `this link <https://en.wikipedia.org/wiki/Linear_elasticity#(An)isotropic_(in)homogeneous_media>`__
for more information about the theory.

Free energy for a linear material is defined as

.. math::

    W(\nabla\mathbf{u}) = 
    \frac{1}{2}\nabla\mathbf{u}\cdot\mathbb{C}\,\nabla\mathbf{u}

For an isotropic material, stress and strain are related through

.. math::

    \mathbb{C}_{ijkl} = \lambda \delta_{ij}\varepsilon_{kk} + 2\mu\varepsilon_{ij}

where :math:`\lambda` and :math:`\mu` are the Lame constant and shear modulus, respectively.
Users can specify these either through (:code:`lame` and :code:`shear`)
OR (:code:`lambda` and :code:`mu`) OR (:code:`E` and :code:`nu`).

Class methods:

#. :code:`Isotropic()`: 
   Basic constructor. Does nothing, and leaves all values initiated as NAN.
#. :code:`Isotropic(Solid<Set::Sym::Isotropic> base)`
   Basic constructor. Does nothing gut allows for inheritance.
#. :code:`Isotropic(Set::Scalar a_mu, Set::Scalar a_lambda)`
   BAD old-fashioned constructor. Do not use!
#. :code:`~Isotropic()`
   Simple destructor. Don't need to change it.
#. :code:`void Define(Set::Scalar a_mu, Set::Scalar a_lambda)`
   BAD old-fashioned way of doing things. Use :code:`Parse` instead.
#. :code:`Set::Scalar W(const Set::Matrix & gradu) const override`
   Returns elastic free energy density
#. :code:`Set::Matrix DW(const Set::Matrix & gradu) const override`
   Returns first derivative of free energy, the stress tensor
#. :code:`Set::Matrix4<...> DDW(const Set::Matrix & ) const override`
   Returns second derivative of free energy, the modulus tensor
#. :code:`virtual void Print(std::ostream &out) const override`
   Prints the modulus tensor object to output stream (usually the terminal)
#. :code:`static Isotropic Random()`
   Static method that generates a random yet acceptable model.
#. :code:`static Isotropic Zero()`
   Static method that generates a "zero" element (so that there is no effect under addition)
#. :code:`static void Parse(Isotropic & value, IO::ParmParse & pp)`
   Parser where all the IO occurs

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