QUAD4_SHELL — 4-node Mindlin-Reissner flat shell#

The 4-node thin-shell kernel. Four corner nodes × 6 DOFs each = 24 DOFs per element. First-order shear-deformation theory (Mindlin 1951; Reissner 1945) — independent rotational DOFs decoupled from the displacement gradient, allowing the element to capture transverse shear (unlike the rotation-from- displacement Kirchhoff family).

  • MAPDL alias: SHELL181

  • Spec: ELEMENTS.QUAD4_SHELL

Kinematics#

The Mindlin-Reissner displacement field at a through-thickness position \(\zeta \in [-t/2, +t/2]\):

\[\begin{split}\mathbf{u}(\xi, \eta, \zeta) = \mathbf{u}_{\mathrm{m}}(\xi, \eta) + \zeta\, \begin{bmatrix} \theta_y \\ -\theta_x \\ 0 \end{bmatrix}.\end{split}\]

Membrane displacements \(\mathbf{u}_{\mathrm{m}}\) and section rotations \((\theta_x, \theta_y)\) are independent Lagrange interpolants on the four corners — the defining feature that admits transverse shear strain \(\boldsymbol{\gamma}\) as a primary variable.

Integration#

Three integration tracks fold into the 24 × 24 stiffness:

  • Membrane + bending: 2 × 2 Gauss-Legendre on the mid- surface (4 points).

  • Transverse shear: 1 × 1 Gauss (centre point) — Malkus & Hughes’ 1978 selective-reduced integration. Suppresses shear locking on thin shells (Reissner’s \(\xi_3 = 0\) shear strain converges to zero in the Kirchhoff limit; the full 2 × 2 rule penalises the discrete approximation, the 1 × 1 rule does not).

  • Drilling-DOF stabilisation: \(\alpha\, G\, t\, A\) per-node penalty (Allman 1984; Hughes-Brezzi 1989) keeps the local 24 × 24 stiffness non-singular when \(\theta_z\) is free. The default \(\alpha = 10^{-3}\) matches the practitioner choice in Cook §17.3.

Real constants#

  • REAL[0] — \(t\), through-thickness shell thickness.

Verification cross-references#

The MITC4 variant of Bathe & Dvorkin (1985) — locking-free even on warped meshes — is on the roadmap.

Implementation: femorph_solver.elements.quad4_shell.

References#

  • Mindlin, R. D. (1951) “Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates,” J. Appl. Mech. 18, 31–38.

  • Reissner, E. (1945) “The effect of transverse shear deformation on the bending of elastic plates,” J. Appl. Mech. 12, A69–A77.

  • Malkus, D. S. and Hughes, T. J. R. (1978) “Mixed finite element methods — reduced and selective integration techniques: a unification of concepts,” CMAME 15 (1), 63–81 (selective-reduced shear).

  • Allman, D. J. (1984) “A compatible triangular element including vertex rotations for plane elasticity analysis,” Computers & Structures 19 (1–2), 1–8 (drilling DOF).

  • Hughes, T. J. R. and Brezzi, F. (1989) “On drilling degrees of freedom,” CMAME 72 (1), 105–121.

  • Bathe, K.-J. and Dvorkin, E. N. (1985) “A four-node plate bending element based on Mindlin / Reissner plate theory and a mixed interpolation,” IJNME 21 (2), 367–383 (MITC4 — roadmap).

  • Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002) Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, §17 (shells).