dynamics_shell_elements Module

Flat, three-dimensional shell elements with six degrees of freedom per node.

Each element is formulated in a local, planar coordinate system by superposing a membrane (plane-stress) element, a plate-bending element, and a drilling-rotation stabilization term. The resulting local matrices are then transformed into the global coordinate system.

The nodal degrees of freedom are ordered as $$\boldsymbol{u}_i = \begin{bmatrix} u_i & v_i & w_i & \theta_{x,i} & \theta_{y,i} & \theta_{z,i} \end{bmatrix}^T,$$ where the rotations follow the right-hand rule about each axis. The element displacement vector stacks the nodal vectors in local node order.

Local Coordinate System

The local $z$-axis is the unit normal of the element plane computed from the node ordering (right-hand rule). The local $x$-axis is the projection of the vector from node 1 to node 2 onto the element plane, and the local $y$-axis completes the right-handed system. Nodes are assumed to lie in a common plane; warped quadrilaterals are projected onto their mean plane.

Kinematics

Within the local system, the displacement through the thickness is $u = u_0 + z\theta_y$ and $v = v_0 - z\theta_x$. The generalized strain vector is $$\boldsymbol{\varepsilon} = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{yy} & \gamma_{xy} & \kappa_{xx} & \kappa_{yy} & \kappa_{xy} & \gamma_{xz} & \gamma_{yz} \end{bmatrix}^T,$$ with $$\kappa_{xx} = \frac{\partial \theta_y}{\partial x}, \quad \kappa_{yy} = -\frac{\partial \theta_x}{\partial y}, \quad \kappa_{xy} = \frac{\partial \theta_y}{\partial y} - \frac{\partial \theta_x}{\partial x},$$ $$\gamma_{xz} = \frac{\partial w}{\partial x} + \theta_y, \quad \gamma_{yz} = \frac{\partial w}{\partial y} - \theta_x.$$

The corresponding stress-resultant vector is $$\boldsymbol{\sigma} = \begin{bmatrix} N_{xx} & N_{yy} & N_{xy} & M_{xx} & M_{yy} & M_{xy} & Q_x & Q_y \end{bmatrix}^T = D \boldsymbol{\varepsilon},$$ where $N$ are membrane forces per unit length, $M$ are moments per unit length, and $Q$ are transverse shear forces per unit length.

Drilling Rotation

The in-plane rotation $\theta_z$ is not part of classical shell theory. It is stabilized with the Hughes-Brezzi penalty $$\Pi_d = \frac{1}{2}\int_A \alpha G t \left( \theta_z - \frac{1}{2}\left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \right)^2 dA,$$ which vanishes for rigid-body motion and prevents a singular stiffness matrix for co-planar meshes. The factor $\alpha$ is the element's drilling_factor.

Mass Matrix

The consistent mass matrix is computed from the linear (triangle) or bilinear (quadrilateral) interpolation of all six nodal quantities with translational inertia $\rho t$ and rotary inertia $\rho t^3 / 12$. The rotary inertia is also assigned to the drilling rotation to keep the mass matrix positive definite.

Contents

Interfaces

rectangular_shell_element triangular_shell_element

Derived Types

rectangular_shell_element shell_element triangular_shell_element

Interfaces

public interface rectangular_shell_element

• private pure function quad_init(mat, thickness, nd1, nd2, nd3, nd4) result(rst)

  Constructs a new [[rectangular_shell_element]].

  Arguments

  Type	Intent	Optional	Attributes		Name
  class(material),	intent(in)			::	mat	The material.
  real(kind=real64),	intent(in)			::	thickness	The shell thickness. This value must be positive.
  class(node),	intent(in)			::	nd1	The first node.
  class(node),	intent(in)			::	nd2	The second node.
  class(node),	intent(in)			::	nd3	The third node.
  class(node),	intent(in)			::	nd4	The fourth node.

  Return Value type(rectangular_shell_element)

  The new [[rectangular_shell_element]].

public interface triangular_shell_element

• private pure function tri_init(mat, thickness, nd1, nd2, nd3) result(rst)

  Constructs a new [[triangular_shell_element]].

  Arguments

  Type	Intent	Optional	Attributes		Name
  class(material),	intent(in)			::	mat	The material.
  real(kind=real64),	intent(in)			::	thickness	The shell thickness. This value must be positive.
  class(node),	intent(in)			::	nd1	The first node.
  class(node),	intent(in)			::	nd2	The second node.
  class(node),	intent(in)			::	nd3	The third node.

  Return Value type(triangular_shell_element)

  The new [[triangular_shell_element]].

Derived Types

type, public, extends(shell_element) ::  rectangular_shell_element

Defines a four-node flat shell element.

Read more…

Components

Type	Visibility	Attributes		Name		Initial
real(kind=real64),	public		::	drilling_factor	=	1.0d-3	The nondimensional drilling-rotation penalty factor $\alpha$. The drilling stiffness is $\alpha G t$, where $G$ is the shear modulus.
type(material),	public		::	material			The material.
type(node),	public,	dimension(4)	::	nodes			The element nodes.
real(kind=real64),	public		::	shear_correction	=	5.0d0/6.0d0	The transverse shear correction factor. This value is only used by elements that include transverse shear deformation.
real(kind=real64),	public		::	thickness			The shell thickness.

Constructor

private pure function quad_init (mat, thickness, nd1, nd2, nd3, nd4)

Constructs a new [[rectangular_shell_element]].

Type-Bound Procedures

procedure , public :: area => shl_area Function
procedure , public :: constitutive_matrix => shl_constitutive_matrix Function
procedure , public :: evaluate_shape_function => quad_shape_function Function
procedure , public :: external_force_vector => shl_ext_force_vector Function
procedure , public :: get_dimensionality => shl_dimensionality Function
procedure , public :: get_dof_per_node => shl_dof_per_node Function
procedure , public :: get_node => quad_get_node Function
procedure , public :: get_node_count => quad_get_node_count Function
procedure , public :: get_node_natural_coordinates => quad_get_node_natural_coordinates Function
procedure , public :: integration_rule => quad_integration_rule Subroutine
procedure , public :: jacobian => shl_jacobian Function
procedure , public :: local_coordinates => shl_local_coordinates Function
procedure , public :: local_frame => shl_local_frame Function
procedure , public :: mass_matrix => shl_mass_matrix Function
procedure , public :: rotation_matrix => shl_rotation_matrix Function
procedure , public :: shape_function_gradient => shl_shape_function_gradient Function
procedure , public :: shape_function_matrix => shl_shape_function_matrix Function
procedure , public :: shape_function_natural_gradient => quad_shape_function_natural_gradient Function
procedure , public :: stiffness_matrix => shl_stiffness_matrix Function
procedure , public :: strain => shl_strain Function
procedure , public :: strain_displacement_matrix => quad_strain_disp_matrix Function
procedure , public :: stress => shl_stress Function

type, public, extends(element) ::  shell_element

Defines a flat shell element with six degrees of freedom per node (three translations and three rotations).

Components

Type	Visibility	Attributes		Name		Initial
real(kind=real64),	public		::	drilling_factor	=	1.0d-3	The nondimensional drilling-rotation penalty factor $\alpha$. The drilling stiffness is $\alpha G t$, where $G$ is the shear modulus.
type(material),	public		::	material			The material.
real(kind=real64),	public		::	shear_correction	=	5.0d0/6.0d0	The transverse shear correction factor. This value is only used by elements that include transverse shear deformation.
real(kind=real64),	public		::	thickness			The shell thickness.

Type-Bound Procedures

procedure , public :: area => shl_area Function
procedure , public :: constitutive_matrix => shl_constitutive_matrix Function
procedure (element_shape_function) , public , pass :: evaluate_shape_function
procedure , public :: external_force_vector => shl_ext_force_vector Function
procedure , public :: get_dimensionality => shl_dimensionality Function
procedure , public :: get_dof_per_node => shl_dof_per_node Function
procedure (element_get_node) , public , pass :: get_node
procedure (element_query) , public , pass :: get_node_count
procedure (element_get_node_natural_coordinates) , public , pass :: get_node_natural_coordinates
procedure (shell_integration_rule) , public , pass :: integration_rule
procedure , public :: jacobian => shl_jacobian Function
procedure , public :: local_coordinates => shl_local_coordinates Function
procedure , public :: local_frame => shl_local_frame Function
procedure , public :: mass_matrix => shl_mass_matrix Function
procedure , public :: rotation_matrix => shl_rotation_matrix Function
procedure , public :: shape_function_gradient => shl_shape_function_gradient Function
procedure , public :: shape_function_matrix => shl_shape_function_matrix Function
procedure (shell_natural_gradient) , public , pass :: shape_function_natural_gradient
procedure , public :: stiffness_matrix => shl_stiffness_matrix Function
procedure , public :: strain => shl_strain Function
procedure (element_matrix_function) , public , pass :: strain_displacement_matrix
procedure , public :: stress => shl_stress Function

type, public, extends(shell_element) ::  triangular_shell_element

Defines a three-node flat shell element.

Read more…

Components

Type	Visibility	Attributes		Name		Initial
real(kind=real64),	public		::	drilling_factor	=	1.0d-3	The nondimensional drilling-rotation penalty factor $\alpha$. The drilling stiffness is $\alpha G t$, where $G$ is the shear modulus.
type(material),	public		::	material			The material.
type(node),	public,	dimension(3)	::	nodes			The element nodes.
real(kind=real64),	public		::	shear_correction	=	5.0d0/6.0d0	The transverse shear correction factor. This value is only used by elements that include transverse shear deformation.
real(kind=real64),	public		::	thickness			The shell thickness.

Constructor

private pure function tri_init (mat, thickness, nd1, nd2, nd3)

Constructs a new [[triangular_shell_element]].

Type-Bound Procedures

procedure , public :: area => shl_area Function
procedure , public :: constitutive_matrix => shl_constitutive_matrix Function
procedure , public :: evaluate_shape_function => tri_shape_function Function
procedure , public :: external_force_vector => shl_ext_force_vector Function
procedure , public :: get_dimensionality => shl_dimensionality Function
procedure , public :: get_dof_per_node => shl_dof_per_node Function
procedure , public :: get_node => tri_get_node Function
procedure , public :: get_node_count => tri_get_node_count Function
procedure , public :: get_node_natural_coordinates => tri_get_node_natural_coordinates Function
procedure , public :: integration_rule => tri_integration_rule Subroutine
procedure , public :: jacobian => shl_jacobian Function
procedure , public :: local_coordinates => shl_local_coordinates Function
procedure , public :: local_frame => shl_local_frame Function
procedure , public :: mass_matrix => shl_mass_matrix Function
procedure , public :: rotation_matrix => shl_rotation_matrix Function
procedure , public :: shape_function_gradient => shl_shape_function_gradient Function
procedure , public :: shape_function_matrix => shl_shape_function_matrix Function
procedure , public :: shape_function_natural_gradient => tri_shape_function_natural_gradient Function
procedure , public :: stiffness_matrix => shl_stiffness_matrix Function
procedure , public :: strain => shl_strain Function
procedure , public :: strain_displacement_matrix => tri_strain_disp_matrix Function
procedure , public :: stress => shl_stress Function