# General stiffness¶

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

## Definition¶

Source and target interfaces can be chosen arbitrarily.

## Parameters¶

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

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

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

A list of symbols is shown in following table.

Symbol

Dimension

Meaning

$$\mathbf{F_T}$$

$$\in\mathbb{R}^{6\times 1}$$

$$\mathbf{F_S}$$

$$\in\mathbb{R}^{6\times 1}$$

$$\mathbf{y_T}$$

$$\in\mathbb{R}^{6\times 1}$$

Displacement vector of target interface

$$\mathbf{y_S}$$

$$\in\mathbb{R}^{6\times 1}$$

Displacement vector of source interface

$$\mathbf{K_{TT}}$$

$$\in\mathbb{R}^{6\times 6}$$

Stiffness matrix coupling target displacement and target load

$$\mathbf{K_{ST}}$$

$$\in\mathbb{R}^{6\times 6}$$

Stiffness matrix coupling target displacement and source load

$$\mathbf{K_{TS}}$$

$$\in\mathbb{R}^{6\times 6}$$

Stiffness matrix coupling source displacement and target load

$$\mathbf{K_{SS}}$$

$$\in\mathbb{R}^{6\times 6}$$

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 $$\mathbf{K}$$ and applied to the coupling equation as follows:

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