fenics_shells.common package

Submodules

fenics_shells.common.constitutive_models module

fenics_shells.common.constitutive_models.psi_M(k, **kwargs)

Returns bending moment energy density calculated from the curvature k using:

Isotropic case: .. math:: D = frac{E*t^3}{24(1 - nu^2)} W_m(k, ldots) = D*((1 - nu)*tr(k**2) + nu*(tr(k))**2)

Parameters:
  • k – Curvature, typically UFL form with shape (2,2) (tensor).
  • **kwargs – Isotropic case: E: Young’s modulus, Constant or Expression. nu: Poisson’s ratio, Constant or Expression. t: Thickness, Constant or Expression.
Returns:

UFL form of bending stress tensor with shape (2,2) (tensor).
fenics_shells.common.constitutive_models.psi_N(e, **kwargs)

Returns membrane energy density calculated from e using:

Isotropic case: .. math:: B = frac{E*t}{2(1 - nu^2)} N(e, ldots) = B(1 - nu)e + nu mathrm{tr}(e)I

Parameters:
  • e – Membrane strain, typically UFL form with shape (2,2) (tensor).
  • **kwargs – Isotropic case: E: Young’s modulus, Constant or Expression. nu: Poisson’s ratio, Constant or Expression. t: Thickness, Constant or Expression.
Returns:

UFL form of membrane stress tensor with shape (2,2) (tensor).
fenics_shells.common.constitutive_models.strain_from_voigt(e_voigt)

Inverse operation of strain_to_voigt.

Parameters:
  • sigma_voigt – UFL form with shape (3,1) corresponding to the strain
  • in Voigt format (pseudo-vector) –
Returns:

a symmetric stress tensor, typically UFL form with shape (2,2)
fenics_shells.common.constitutive_models.strain_to_voigt(e)

Returns the pseudo-vector in the Voigt notation associate to a 2x2 symmetric strain tensor, according to the following rule (see e.g. https://en.wikipedia.org/wiki/Voigt_notation),

\[\begin{split}e = \begin{bmatrix} e_{00} & e_{01}\\ e_{01} & e_{11} \end{bmatrix}\quad\to\quad e_\mathrm{voigt}= \begin{bmatrix} e_{00} & e_{11}& 2e_{01} \end{bmatrix}\end{split}\]
Parameters:e – a symmetric 2x2 strain tensor, typically UFL form with shape (2,2)
Returns:a UFL form with shape (3,1) corresponding to the input tensor in Voigt notation.
fenics_shells.common.constitutive_models.stress_from_voigt(sigma_voigt)

Inverse operation of stress_to_voigt.

Parameters:
  • sigma_voigt – UFL form with shape (3,1) corresponding to the stress
  • in Voigt format. (pseudo-vector) –
Returns:

a symmetric stress tensor, typically UFL form with shape (2,2)
fenics_shells.common.constitutive_models.stress_to_voigt(sigma)

Returns the pseudo-vector in the Voigt notation associate to a 2x2 symmetric stress tensor, according to the following rule (see e.g. https://en.wikipedia.org/wiki/Voigt_notation),

\[\begin{split}\sigma = \begin{bmatrix} \sigma_{00} & \sigma_{01}\\ \sigma_{01} & \sigma_{11} \end{bmatrix}\quad\to\quad \sigma_\mathrm{voigt}= \begin{bmatrix} \sigma_{00} & \sigma_{11}& \sigma_{01} \end{bmatrix}\end{split}\]
Parameters:
  • sigma – a symmetric 2x2 stress tensor, typically UFL form with shape
  • (2,2) –
Returns:

a UFL form with shape (3,1) corresponding to the input tensor in Voigt notation.

fenics_shells.common.energy module

fenics_shells.common.energy.membrane_bending_energy(e, k, A, D, B)

Return the coupled membrane-bending energy for a plate model.

Parameters:
  • e – Membrane strains, UFL or DOLFIN Function of rank (2, 2) (tensor).
  • k – Curvature, UFL or DOLFIN Function of rank (2, 2) (tensor).
  • A – Membrane stresses.
  • D – Bending stresses.
  • B – Coupled membrane-bending stresses.
fenics_shells.common.energy.membrane_energy(e, N)

Return internal membrane energy for a plate model.

Parameters:
  • e – Membrane strains, UFL or DOLFIN Function of rank (2, 2) (tensor).
  • N – Membrane stress, UFL or DOLFIN Function of rank (2, 2) (tensor).
Returns:

UFL form of internal elastic membrane energy for a plate model.

fenics_shells.common.kinematics module

fenics_shells.common.kinematics.F(u)

Return deformation gradient tensor for non-linear plate model.

Deformation gradient of 2-dimensional manifold embedded in 3-dimensional space.

\[F = I + \nabla u\]
Parameters:u – displacement field, typically UFL (3,1) coefficient.
Returns:a UFL coeffient with shape (3,2)
fenics_shells.common.kinematics.e(u)

Return membrane strain tensor for linear plate model.

\[e = \dfrac{1}{2}(\nabla u+\nabla u^T)\]
Parameters:u – membrane displacement field, typically UFL (2,1) coefficient.
Returns:a UFL form with shape (2,2)
fenics_shells.common.kinematics.k(theta)

Return bending curvature tensor for linear plate model.

\[k = \dfrac{1}{2}(\nabla \theta+\nabla \theta^T)\]
Parameters:theta – rotation field, typically UFL (2,1) form or a dolfin Function
Returns:a UFL form with shape (2,2)

fenics_shells.common.laminates module

fenics_shells.common.laminates.ABD(E1, E2, G12, nu12, hs, thetas)

Return the stiffness matrix of a kirchhoff-love model of a laminate obtained by stacking n orthotropic laminae with possibly different thinknesses and orientations (see Reddy 1997, eqn 1.3.71).

It assumes a plane-stress state.

Parameters:
  • E1 – The Young modulus in the material direction 1.
  • E2 – The Young modulus in the material direction 2.
  • G12 – The in-plane shear modulus.
  • nu12 – The in-plane Poisson ratio.
  • hs – a list with length n with the thicknesses of the layers (from top to bottom).
  • theta – a list with the n orientations (in radians) of the layers (from top to bottom).
Returns:

a symmetric 3x3 ufl matrix giving the membrane stiffness in Voigt notation. B: a symmetric 3x3 ufl matrix giving the membrane/bending coupling stiffness in Voigt notation. D: a symmetric 3x3 ufl matrix giving the bending stiffness in Voigt notation.
Return type:

A
fenics_shells.common.laminates.F(G13, G23, hs, thetas)

Return the shear stiffness matrix of a Reissner-Midlin model of a laminate obtained by stacking n orthotropic laminae with possibly different thinknesses and orientations. (See Reddy 1997, eqn 3.4.18)

It assumes a plane-stress state.

Parameters:
  • G13 – The transverse shear modulus between the material directions 1-3.
  • G23 – The transverse shear modulus between the material directions 2-3.
  • hs – a list with length n with the thicknesses of the layers (from top to bottom).
  • theta – a list with the n orientations (in radians) of the layers (from top to bottom).
Returns:

a symmetric 2x2 ufl matrix giving the shear stiffness in Voigt notation.
Return type:

F
fenics_shells.common.laminates.NM_T(E1, E2, G12, nu12, hs, thetas, DeltaT_0, DeltaT_1=0.0, alpha1=1.0, alpha2=1.0)

Return the thermal stress and moment resultant of a Kirchhoff-Love model of a laminate obtained by stacking n orthotropic laminae with possibly different thinknesses and orientations.

It assumes a plane-stress states and a temperature distribution in the from

Delta(z) = DeltaT_0 + z * DeltaT_1

Parameters:
  • E1 – The Young modulus in the material direction 1.
  • E2 – The Young modulus in the material direction 2.
  • G12 – The in-plane shear modulus.
  • nu12 – The in-plane Poisson ratio.
  • hs – a list with length n with the thicknesses of the layers (from top to bottom).
  • theta – a list with the n orientations (in radians) of the layers (from top to bottom).
  • alpha1 – Expansion coefficient in the material direction 1.
  • alpha2 – Expansion coefficient in the material direction 2.
  • DeltaT_0 – Average temperature field.
  • DeltaT_1 – Gradient of the temperature field.
Returns:

a 3x1 ufl vector giving the membrane inelastic stress. M_T: a 3x1 ufl vector giving the bending inelastic stress.
Return type:

N_T
fenics_shells.common.laminates.rotated_lamina_expansion_inplane(alpha11, alpha22, theta)

Return the in-plane expansion matrix of an orhtropic layer in a reference rotated by an angle theta wrt to the material one. It assumes Voigt notation and plane stress state. (See Reddy 1997, eqn 1.3.71)

Parameters:
  • alpha11 – Expansion coefficient in the material direction 1.
  • alpha22 – Expansion coefficient in the material direction 2.
  • theta – The rotation angle from the material to the desired reference system.
Returns:

a 3x1 ufl vector giving the expansion matrix in voigt notation.
Return type:

alpha_theta
fenics_shells.common.laminates.rotated_lamina_stiffness_inplane(E1, E2, G12, nu12, theta)

Return the in-plane stiffness matrix of an orhtropic layer in a reference rotated by an angle theta wrt to the material one. It assumes Voigt notation and plane stress state. (See Reddy 1997, eqn 1.3.71)

Parameters:
  • E1 – The Young modulus in the material direction 1.
  • E2 – The Young modulus in the material direction 2.
  • G23 – The in-plane shear modulus
  • nu12 – The in-plane Poisson ratio
  • theta – The rotation angle from the material to the desired refence system
Returns:

a 3x3 symmetric ufl matrix giving the stiffness matrix
Return type:

Q_theta
fenics_shells.common.laminates.rotated_lamina_stiffness_shear(G13, G23, theta, kappa=0.8333333333333334)

Return the shear stiffness matrix of an orhtropic layer in a reference rotated by an angle theta wrt to the material one. It assumes Voigt notation and plane stress state (see Reddy 1997, eqn 3.4.18).

Parameters:
  • G12 – The transverse shear modulus between the material directions 1-2.
  • G13 – The transverse shear modulus between the material directions 1-3.
  • kappa – The shear correction factor.
Returns:

a 3x3 symmetric ufl matrix giving the stiffness matrix.
Return type:

Q_shear_theta
fenics_shells.common.laminates.z_coordinates(hs)

Return a list with the thickness coordinate of the top surface of each layer taking the midplane as z = 0.

Parameters:hs – a list giving the thinckesses of each layer ordered from bottom (layer - 0) to top (layer n-1).
Returns:
a list of coordinate of the top surface of each layer
ordered from bottom (layer - 0) to top (layer n-1)
Return type:z

Module contents