QUAD4_PLANE — 4-node bilinear plane quad
========================================

The 2D bilinear quadrilateral, used for plane-stress and
plane-strain analysis.  Four corner nodes × 2 in-plane
translations = 8 DOFs per element.

* **Spec:** ``ELEMENTS.QUAD4_PLANE``

Shape functions
---------------

Bilinear Lagrange on the reference square
:math:`\hat\Omega = [-1, +1]^{2}` with corner-node sign vector
:math:`(\xi_i, \eta_i) \in \{-1, +1\}^{2}`:

.. math::

    N_i(\xi, \eta) =
    \tfrac{1}{4}(1 + \xi_i\, \xi)(1 + \eta_i\, \eta).

These satisfy Kronecker-delta interpolation at the four
corners and partition-of-unity on :math:`\hat\Omega`.  See
:ref:`sphx_glr_gallery_elements_plane182_example_quad4_shape_functions.py`
for contour plots and a verified basis-matrix identity at every
corner.

Integration
-----------

* **2 × 2 Gauss-Legendre** for both stiffness and mass.
  Exact for the bilinear strain-displacement product on a
  regular quad — see :doc:`../theory/quadrature`.

Constitutive
------------

Plane-stress vs plane-strain selected by ``ELEMENTS.QUAD4_PLANE(mode=...)``:

* ``"stress"`` (default) — out-of-plane normal stress
  :math:`\sigma_{zz} = 0`.  3 × 3 elastic matrix:

  .. math::

      \mathbf{C}^{(\sigma)} =
      \frac{E}{1 - \nu^{2}}
      \begin{bmatrix}
          1 & \nu & 0 \\
          \nu & 1 & 0 \\
          0 & 0 & (1 - \nu)/2
      \end{bmatrix}.

* ``"strain"`` — out-of-plane strain :math:`\varepsilon_{zz} = 0`.
  3 × 3 elastic matrix:

  .. math::

      \mathbf{C}^{(\varepsilon)} =
      \frac{E}{(1 + \nu)(1 - 2\nu)}
      \begin{bmatrix}
          1 - \nu & \nu & 0 \\
          \nu & 1 - \nu & 0 \\
          0 & 0 & (1 - 2\nu)/2
      \end{bmatrix}.

Real constants
--------------

* ``REAL[0]`` — :math:`t`, out-of-plane thickness (used to
  scale the plane-stress integral; ignored under plane strain).

Stress recovery
---------------

The strain-displacement matrix evaluated at each Gauss /
node point gives 3-component Voigt strain
:math:`(\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy})`;
:func:`compute_nodal_stress
<femorph_solver.result._stress_recovery.compute_nodal_stress>`
applies the 3 × 3 :math:`\mathbf{C}` matrix above to recover
in-plane stress.  Issue #262 wired the plane-element strain
extraction into the canonical recovery path
(``eel_batch`` returns 3-component strain, padded to 6-Voigt
for the global stress-averaging machinery).

Verification cross-references
-----------------------------

* :ref:`sphx_glr_gallery_elements_plane182_example_plane182.py` — single-quad uniaxial-tension test.
* :ref:`sphx_glr_gallery_verification_example_verify_nafems_le1.py` — NAFEMS LE1 elliptic membrane (stress-concentration benchmark, σ_yy(D) = 92.7 MPa).

Implementation: :mod:`femorph_solver.elements.quad4_plane`.

References
----------

* Hughes, T. J. R. (2000) *The Finite Element Method —
  Linear Static and Dynamic Finite Element Analysis*, Dover,
  §3.5 + §3.6 (bilinear quad).
* Zienkiewicz, O. C. and Taylor, R. L. (2013) *The Finite
  Element Method: Its Basis and Fundamentals*, 7th ed.,
  §6.3.2 + §6.4.
* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J.
  (2002) *Concepts and Applications of Finite Element
  Analysis*, 4th ed., Wiley, §3.5 (plane stress), §3.6
  (plane strain).