Axisymmetric Elements ===================== Axisymmetric Elements --------------------- Implementation relies on elements located exclusively in :math:`x,y` plane. The coordinate :math:`x` corresponds to radius, :math:`y` is the axis of rotation. Approximation of displacement functions :math:`u,v` is carried out on a particular finite element. Nonzero strains read .. math:: \varepsilon_{x}=\varepsilon_{r}&=&\frac{\partial u}{\partial x}\\ \varepsilon_{y}=\varepsilon_{z}&=&\frac{\partial v}{\partial y}\\ \varepsilon_{\theta}&=&\frac{u}{r}\\ \gamma_{xy}=\gamma_{rz}&=&\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} Stress components can be computed from elasticity matrix. Note that this matrix corresponds to a submatrix of the full 3D elasticity matrix. .. math:: \left\{\begin{array}{c} \sigma_x \\ \sigma_y \\ \sigma_\theta \\ \sigma_{xy} \end{array} \right\} = \frac{E}{(1+\nu)(1-2\nu)}\left[ \begin{array}{cccc} 1-\nu & \nu & \nu & 0 \\ \nu & 1-\nu & \nu & 0 \\ \nu & \nu & 1-\nu & 0 \\ 0 & 0 & 0 & (1-2\nu)/2 \end{array} \right] = \left\{\begin{array}{c} \varepsilon_x \\ \varepsilon_y \\ \varepsilon_\theta \\ \gamma_{xy} \end{array} \right\} In OOFEM, the strain vector is arranged as :math:`\{\varepsilon_{x}, \varepsilon_{y}, \varepsilon_{\theta}, 0, 0, \gamma_{xy}\}^T` and the stress vector :math:`\{\sigma_{x}, \sigma_{y}, \sigma_{\theta}, 0, 0, \tau_{xy}\}^T`. Implementation assumes a segment of 1 rad. Axisymm3d element ~~~~~~~~~~~~~~~~~ Implementation of triangular three-node finite element for axisymmetric continuum. Each node has 2 degrees of freedom. Node numbering and edge position is the same as in Fig. :ref:`TrPlanestressfig`. The element features are summarized in Table :numref:`axisymm3dsummary`. .. table:: Axisymm3d element summary :name: axisymm3dsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | Axisymm3d | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | Triangular axisymmetric linear element | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` :optelemparam:`NIPfish{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`NIP`: allows to set the number of integration points (possible completions are 1 | | | (default), 4 and 7 point integration rule). | +--------------------------+----------------------------------------------------------------------------------------------+ | Unknowns | Two dofs (u-displacement, v-displacement) are required in each node. | +--------------------------+----------------------------------------------------------------------------------------------+ | Approximation | Linear approximation of displacement and geometry. | +--------------------------+----------------------------------------------------------------------------------------------+ | Integration | The integration can be altered using :param:`NIP` paramter (default is 1 point integration). | +--------------------------+----------------------------------------------------------------------------------------------+ | Features | - | +--------------------------+----------------------------------------------------------------------------------------------+ | CS properties | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Loads | Boundary and body loads are supported. | +--------------------------+----------------------------------------------------------------------------------------------+ | Nlgeo | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Tests/Examples | `sm/axisymm06.in | | | `_, | | | `sm/axisymm04.in | | | `_, | | | `sm/axisymm05.in | | | `_ | +--------------------------+----------------------------------------------------------------------------------------------+ Q4axisymm element ~~~~~~~~~~~~~~~~~ Implementation of quadratic isoparametric eight-node quadrilateral - finite element for axisymmetric 3d continuum. Each node has 2 degrees of freedom. The element features are summarized in :numref:`q4axisymmsummary`. .. table:: Q4axisymm element summary :name: q4axisymmsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | Q4axisymm | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | Quadratic isoparametric eight-node quadrilateral for axisymmetric analysis | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` :optelemparam:`NIPfish{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`NIP`: allows to set the number of integration points for integration of terms | | | corresponding to :math:`\varepsilon_x` and :math:`\varepsilon_y` strains (possible | | | completions are 1, 4 (default), 9, and 16). :param:`NIPfish`: allows to set the number of | | | integration points for integration of remain terms (corresponding to | | | :math:`\varepsilon_\theta` and :math:`\gamma_{rz}`) (Supported values include 1 (default), | | | 4, 9, and 16 integration point formula). | +--------------------------+----------------------------------------------------------------------------------------------+ | Unknowns | Two dofs (u-displacement, v-displacement) are required in each node. | +--------------------------+----------------------------------------------------------------------------------------------+ | Approximation | Quadratic approximation of displacement and geometry. | +--------------------------+----------------------------------------------------------------------------------------------+ | Integration | The integration of terms corresponding to :math:`\varepsilon_x` and :math:`\varepsilon_y` | | | strains can be altered using :param:`NIP` parameter (default is 4 point formula). The | | | remaining terms (creesponding to :math:`\varepsilon_\theta` and :math:`\gamma_{rz}`) are | | | integrated by default using 1 point formula (see :param:`NIPfish` parameter). | +--------------------------+----------------------------------------------------------------------------------------------+ | Features | - | +--------------------------+----------------------------------------------------------------------------------------------+ | CS properties | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Loads | No boundary and body loads are supported. | +--------------------------+----------------------------------------------------------------------------------------------+ | Nlgeo | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+ L4axisymm element ~~~~~~~~~~~~~~~~~ Implementation of isoparametric four-node quadrilateral axisymmetric finite element with linear interpolations of displacements :math:`u, v`. Node numbering and edge position is the same as in Fig. :ref:`Planestress2dfig`. The element features are summarized in :numref:`l4axisymmsummary`. .. table:: L4axisymm element summary :name: l4axisymmsummary +--------------------------+----------------------------------------------------------------------------------------------+ | Keyword | L4axisymm | +--------------------------+----------------------------------------------------------------------------------------------+ | Description | Isoparametric four-node quadrilateral element for axisymmetric analysis | +--------------------------+----------------------------------------------------------------------------------------------+ | Specific parameters | :optelemparam:`NIP{in}` | +--------------------------+----------------------------------------------------------------------------------------------+ | Parameters | :param:`NIP`: allows to set the number of integration points for integration of terms | | | corresponding to :math:`\varepsilon_x` and :math:`\varepsilon_y` strains (possible | | | completions are 1, 4 (default), 9, and 16). | +--------------------------+----------------------------------------------------------------------------------------------+ | Unknowns | Two dofs (u-displacement, v-displacement) are required in each node. | +--------------------------+----------------------------------------------------------------------------------------------+ | Approximation | Linear approximation of displacement and geometry. | +--------------------------+----------------------------------------------------------------------------------------------+ | Integration | The integration of :math:`\varepsilon_x` and :math:`\varepsilon_y` strains can be altered | | | using :param:`NIP` parameter (possible completions are 1, 4 (default), 9 or 16 point | | | integration rule). The remaining strain components (:math:`\varepsilon_\theta` and | | | :math:`\gamma_{rz}`) are integrated using one point integration formula. | +--------------------------+----------------------------------------------------------------------------------------------+ | Features | - | +--------------------------+----------------------------------------------------------------------------------------------+ | CS properties | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Loads | Boundary and body loads supported. | +--------------------------+----------------------------------------------------------------------------------------------+ | Nlgeo | 0. | +--------------------------+----------------------------------------------------------------------------------------------+ | Status | - | +--------------------------+----------------------------------------------------------------------------------------------+ | Tests/Examples | `sm/axisymm03.in | | | `_, | | | `sm/axisymm01.in | | | `_, | | | `sm/axisymm02.in | | | `_ | +--------------------------+----------------------------------------------------------------------------------------------+