FEniCS-Shells Release 2018.1.0 Aug 02, 2021 Contents 1 Subpackages 1 2 Module contents 13 3 Documented demos 15 4 FEniCS-Shells 61 Bibliography 65 Python Module Index 67 Index 69 i ii CHAPTER 1 Subpackages 1.1 fenics_shells.analytical package 1.1.1 Submodules 1.1.2 fenics_shells.analytical.lovadina_clamped module Analytical solution for clamped Reissner-Mindlin plate problem from Lovadina et al. 1.1.3 fenics_shells.analytical.simply_supported module Analytical solution for simply-supported Reissner-Mindlin square plate under a uniform transverse load. 1.1.4 fenics_shells.analytical.vonkarman_heated module Analytical solution for elliptic orthotropic von Karman plate with lenticular thickness subject to a uniform field of inelastic curvatures. fenics_shells.analytical.vonkarman_heated.analytical_solution(Ai, Di, a_rad, b_rad) 1 FEniCS-Shells, Release 2018.1.0 1.1.5 Module contents 1.2 fenics_shells.common package 1.2.1 Submodules 1.2.2 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), [︂ ]︂ e00 e01 [︀ ]︀ e = ! evoigt = e00 e11 2e01 e01 e11 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. 2 Chapter 1. Subpackages FEniCS-Shells, Release 2018.1.0 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), [︂ ]︂ 휎00 휎01 [︀ ]︀ 휎 = ! 휎voigt = 휎00 휎11 휎01 휎01 휎11 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. 1.2.3 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. 1.2.4 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 + ru Parameters u – displacement field, typically UFL (3,1) coefficient. 1.2. fenics_shells.common package 3 FEniCS-Shells, Release 2018.1.0 Returns a UFL coeffient with shape (3,2) fenics_shells.common.kinematics.e(u) Return membrane strain tensor for linear plate model. 1 e = (ru + ruT ) 2 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. 1 k = (r휃 + r휃T ) 2 Parameters theta – rotation field, typically UFL (2,1) form or a dolfin Function Returns a UFL form with shape (2,2) 1.2.5 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). 4 Chapter 1. Subpackages FEniCS-Shells, Release 2018.1.0 • 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, al- pha22, 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. 1.2. fenics_shells.common package 5 FEniCS-Shells, Release 2018.1.0 • 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).