sfepy.mechanics.membranes module

sfepy.mechanics.membranes.create_mapping(coors, gel, order)[source]

Create mapping from transformed (in x-y plane) element faces to reference element faces.

Parameters:

coors : array

The transformed coordinates of element nodes, shape (n_el, n_ep, dim). The function verifies that the all z components are zero.

gel : GeometryElement instance

The geometry element corresponding to the faces.

order : int

The polynomial order of the mapping.
Returns:

mapping : VolumeMapping instance

The reference element face mapping.
sfepy.mechanics.membranes.create_transformation_matrix(coors)[source]

Create a transposed coordinate transformation matrix, that transforms 3D coordinates of element face nodes so that the transformed nodes are in the x-y plane. The rotation is performed w.r.t. the first node of each face.

Parameters:

coors : array

The coordinates of element nodes, shape (n_el, n_ep, dim).
Returns:

mtx_t : array

The transposed transformation matrix T, i.e. X_{inplane} = T^T X_{3D}.

Notes

T = [t_1, t_2, n], where t_1, t_2, are unit in-plane (column) vectors and n is the unit normal vector, all mutually orthonormal.

sfepy.mechanics.membranes.describe_deformation(el_disps, bfg)[source]

Describe deformation of a thin incompressible 2D membrane in 3D space, composed of flat finite element faces.

The coordinate system of each element (face), i.e. the membrane mid-surface, should coincide with the x, y axes of the x-y plane.

Parameters:

el_disps : array

The displacements of element nodes, shape (n_el, n_ep, dim).

bfg : array

The in-plane base function gradients, shape (n_el, n_qp, dim-1, n_ep).
Returns:

mtx_c ; array

The in-plane right Cauchy-Green deformation tensor C_{ij}, i, j = 1, 2.

c33 : array

The component C_{33} computed from the incompressibility condition.

mtx_b : array

The discrete Green strain variation operator.
sfepy.mechanics.membranes.describe_geometry(ig, field, region, integral)[source]

Describe membrane geometry in a given region.

Parameters:

ig : int

The element group index.

field : Field instance

The field defining the FE approximation.

region : Region instance

The surface region to describe.

integral : Integral instance

The integral defining the quadrature points.
Returns:

mtx_t : array

The transposed transformation matrix T, see create_transformation_matrix().

membrane_geo : CMapping instance

The mapping from transformed elements to a reference elements.
sfepy.mechanics.membranes.get_green_strain_sym3d(mtx_c, c33)[source]

Get the 3D Green strain tensor in symmetric storage.

Parameters:

mtx_c ; array

The in-plane right Cauchy-Green deformation tensor C_{ij}, i, j = 1, 2, shape (n_el, n_qp, dim-1, dim-1).

c33 : array

The component C_{33} computed from the incompressibility condition, shape (n_el, n_qp).
Returns:

mtx_e : array

The membrane Green strain E_{ij} = \frac{1}{2} (C_{ij}) - \delta_{ij}, symmetric storage: items (11, 22, 33, 12, 13, 23), shape (n_el, n_qp, sym, 1).
sfepy.mechanics.membranes.get_invariants(mtx_c, c33)[source]

Get the first and second invariants of the right Cauchy-Green deformation tensor describing deformation of an incompressible membrane.

Parameters:

mtx_c ; array

The in-plane right Cauchy-Green deformation tensor C_{ij}, i, j = 1, 2, shape (n_el, n_qp, dim-1, dim-1).

c33 : array

The component C_{33} computed from the incompressibility condition, shape (n_el, n_qp).
Returns:

i1 : array

The first invariant of C_{ij}.

i2 : array

The second invariant of C_{ij}.
sfepy.mechanics.membranes.get_tangent_stress_matrix(stress, bfg)[source]

Get the tangent stress matrix of a thin incompressible 2D membrane in 3D space, given a stress.

Parameters:

stress : array

The components 11, 22, 12 of the second Piola-Kirchhoff stress tensor, shape (n_el, n_qp, 3, 1).

bfg : array

The in-plane base function gradients, shape (n_el, n_qp, dim-1, n_ep).
Returns:

mtx : array

The tangent stress matrix, shape (n_el, n_qp, dim*n_ep, dim*n_ep).