6.2 – Matrizenformulierung

 

Auszug aus dem Skript der Vorlesung Finite Elemente bei Dr.-Ing Philipp Höfer an der UniBw München.

Spaltenmatrix der Spannungen:

{\left\{ \sigma \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{\sigma _{xx}}}&{{\sigma _{yy}}}&{{\sigma _{zz}}}&{{\sigma _{xy}}}&{{\sigma _{yz}}}&{{\sigma _{zx}}} \end{array}} \right\}

Spaltenmatrix der Verzerrungen:

{\left\{ \varepsilon \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{\varepsilon _{xx}}}&{{\varepsilon _{yy}}}&{{\varepsilon _{zz}}}&{{\gamma _{xy}}}&{{\gamma _{yz}}}&{{\gamma _{zx}}} \end{array}} \right\},\quad \quad \quad {\gamma _{jk}}\mathop = \limits^{j \ne k} 2{\varepsilon _{jk}}

{\left\{{\delta \varepsilon } \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\delta {\varepsilon _{xx}}}&{\delta {\varepsilon _{yy}}}&{\delta {\varepsilon _{zz}}}&{\delta {\gamma _{xy}}}&{\delta {\gamma _{yz}}}&{\delta {\gamma _{zx}}} \end{array}} \right\}

Spaltenmatrix der Verschiebungen:

{\left\{ u \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{u_x}}&{{u_y}}&{{u_z}} \end{array}} \right\},\quad \quad {\left\{{\delta u} \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\delta {u_x}}&{\delta {u_y}}&{\delta {u_z}} \end{array}} \right\}

Spaltenmatrix der Volumen- und Oberflächenkräfte:

{\left\{ k \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{k_x}}&{{k_y}}&{{k_z}} \end{array}} \right\},\quad \quad \quad {\left\{ s \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{s_x}}&{{s_y}}&{{s_z}} \end{array}} \right\}

Einsetzen in die Arbeit der aktuellen Spannungen an den virtuellen Verzerrungen:

\delta {W_\sigma } = \int\limits_V {\mathbb{T} \cdot \delta \mathbb{E}dV} = \int\limits_V {{\sigma _{jk}}\delta {\varepsilon _{jk}}dV} = \int\limits_V {{{\left\{{\delta \varepsilon } \right\}}^T}\left\{ \sigma \right\}dV}

Anteil der Volumen- und Oberflächenkräfte / Trägheitskräfte an den virtuellen Verschiebungen:

\delta {W_a} = \int\limits_A {\vec s \cdot \delta \vec udA} +\int\limits_V {\rho \vec k \cdot \delta \vec udV} = \int\limits_A {{{\left\{{\delta u} \right\}}^T}\left\{ s \right\}dA} +\int\limits_V {\rho {{\left\{{\delta u} \right\}}^T}\left\{ k \right\}dV}

\delta {W_T} = -\int\limits_V {\rho \ddot \vec u \cdot \delta \vec udV} = -\int\limits_V {\rho {{\left\{{\delta u} \right\}}^T}\left\{{\ddot u} \right\}dV}