=================
General stiffness
=================

The general stiffness property implements a linear elastic stiffness between source and target 
interface.

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Definition
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Source and target interfaces can be chosen arbitrarily.

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Parameters
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General stiffness link properties can be parametrised in two modes: `simple` and `advanced`, which are described in the following.

^^^^^^^^^^^^^
Advanced mode
^^^^^^^^^^^^^

In the advanced mode, the user is prompted to supply four stiffness matrices, :math:`\mathbf{K_{SS}}`, :math:`\mathbf{K_{ST}}`, :math:`\mathbf{K_{TS}}`, and :math:`\mathbf{K_{TT}}`. Following equation describes how the stiffness matrices define a coupling between the target and source interface, denoted by subscript :math:`{T}` and :math:`{S}`, respectively.

.. math::

   \begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{K_{TT}} & \mathbf{K_{TS}} \\ \mathbf{K_{ST}} & \mathbf{K_{SS}}\end{bmatrix} \, \begin{bmatrix} \mathbf{y_T} \\ \mathbf{y_S} \end{bmatrix}

A list of symbols is shown in following table.

=======================  =================================  =============================================================================
Symbol                   Dimension                          Meaning                                                                      
=======================  =================================  =============================================================================
:math:`\mathbf{F_T}`     :math:`\in\mathbb{R}^{6\times 1}`  Load vector to target interface                                              
:math:`\mathbf{F_S}`     :math:`\in\mathbb{R}^{6\times 1}`  Load vector to target interface                                              
:math:`\mathbf{y_T}`     :math:`\in\mathbb{R}^{6\times 1}`  Displacement vector of target interface                                      
:math:`\mathbf{y_S}`     :math:`\in\mathbb{R}^{6\times 1}`  Displacement vector of source interface                                      
:math:`\mathbf{K_{TT}}`  :math:`\in\mathbb{R}^{6\times 6}`  Stiffness matrix coupling target displacement and target load                
:math:`\mathbf{K_{ST}}`  :math:`\in\mathbb{R}^{6\times 6}`  Stiffness matrix coupling target displacement and source load                
:math:`\mathbf{K_{TS}}`  :math:`\in\mathbb{R}^{6\times 6}`  Stiffness matrix coupling source displacement and target load                
:math:`\mathbf{K_{SS}}`  :math:`\in\mathbb{R}^{6\times 6}`  Stiffness matrix coupling source displacement and source load                
=======================  =================================  =============================================================================

^^^^^^^^^^^
Simple mode
^^^^^^^^^^^

In the simple mode, a symmetric behaviour between target and source interface is implied. The four stiffness matrices are replaced by a single stiffness matrix :math:`\mathbf{K}` and applied to the coupling equation as follows:
   
.. math::

   \begin{bmatrix}\mathbf{F_T} \\ \mathbf{F_S} \end{bmatrix} = \begin{bmatrix}\mathbf{K} & -\mathbf{K} \\ -\mathbf{K} & \mathbf{K}\end{bmatrix} \, \begin{bmatrix} \mathbf{y_T} \\ \mathbf{y_S} \end{bmatrix}