Single hex — uniaxial tension (Hooke's law)
============================================

The most elementary first-principles correctness check for any
3D continuum element: does a single hex under uniaxial tension
produce the stress and strain a textbook predicts?

For a unit cube of isotropic linear-elastic material with the
``x = 0`` face axially fixed and a uniform axial traction on the
``x = L`` face:

.. math::

    \sigma_{xx} = E \varepsilon_{xx}, \qquad
    \varepsilon_{yy} = \varepsilon_{zz} = -\nu \varepsilon_{xx},
    \qquad \varepsilon_{xy} = \varepsilon_{yz} = \varepsilon_{xz} = 0.

This problem is **exactly** what the element kernel and the
constitutive law encode — if a single hex fails it, something is
wrong at the material-law or strain-recovery layer, not at the
numerical-convergence layer.

Problem
-------

.. list-table::
    :header-rows: 1
    :widths: 30 70

    * - Parameter
      - Value
    * - Geometry
      - Unit cube, ``L × L × L`` with ``L = 1 m``
    * - Element
      - Single ``HEX8`` (``SOLID185``) — one cell
    * - Young's modulus ``E``
      - 200 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Axial load
      - 1 MN distributed on four corner nodes of the ``x = L``
        face (→ uniform nominal stress σ\ :sub:`xx` = 1 MPa)
    * - BCs
      - ``UX = 0`` on every node of the ``x = 0`` face; single-
        node UY / UZ anchors on corners to remove rigid-body
        modes without over-constraining the transverse Poisson
        contraction

Analytical reference
--------------------

Hooke's law in 3D (Cook et al. §1.3; Hughes §2.7 — public-domain
derivations):

.. math::

    \varepsilon_{xx} = \frac{\sigma_{xx}}{E}
                      = \frac{1 \text{ MPa}}{200 \text{ GPa}}
                      = 5.0 \times 10^{-6}

    \varepsilon_{yy} = \varepsilon_{zz} = -\nu \varepsilon_{xx}
                                        = -1.5 \times 10^{-6}

    \sigma_{xx} = 1.0 \times 10^{6} \text{ Pa}, \qquad
    \sigma_{yy} = \sigma_{zz} = \tau_{ij} = 0.

No approximation; no convergence discussion.  The HEX8 shape
functions are linear in each Cartesian direction, so a single
element can represent the uniform strain state *exactly*.

femorph-solver result
---------------------

Ran by :file:`tests/analytical/test_single_hex_uniaxial.py` using
the Model's ``.strain(u)`` on-the-fly API (Agent 1's TA-9b
contribution).  With the strain-recovery path returning
per-node Voigt strain in ``(xx, yy, zz, xy, yz, xz)`` order:

.. list-table::
    :header-rows: 1
    :widths: 25 25 25 25

    * - Component
      - Analytical
      - femorph-solver
      - Relative error
    * - ``ε_xx``
      - ``5.000 × 10⁻⁶``
      - ``5.000 × 10⁻⁶``
      - < 10⁻⁸
    * - ``ε_yy``
      - ``-1.500 × 10⁻⁶``
      - ``-1.500 × 10⁻⁶``
      - < 10⁻⁸
    * - ``ε_zz``
      - ``-1.500 × 10⁻⁶``
      - ``-1.500 × 10⁻⁶``
      - < 10⁻⁸
    * - ``ε_xy``, ``ε_yz``, ``ε_xz``
      - 0
      - < 10⁻¹⁰
      - —

Hooke's law holds to machine precision.  The regression test sets
``rtol = 1 × 10⁻⁸`` for the diagonal components (tight enough to
catch a material-law bug but loose enough for single-precision
element-return-path rounding) and ``1 × 10⁻¹⁰`` absolute for the
shear components.

Cross-references
----------------

.. list-table::
    :header-rows: 1
    :widths: 35 30 35

    * - Source
      - Reported ``ε_xx``
      - Problem ID / location
    * - Closed form (Hooke's law)
      - ``5.00 × 10⁻⁶``
      - Cook *CAFEA* §1.3 + Hughes *FEM* §2.7
    * - NAFEMS *Background to Benchmarks*
      - ``5.00 × 10⁻⁶``
      - BtB-2.1 (uniaxial tension test)
    * - femorph-solver
      - ``5.00 × 10⁻⁶``
      - ``test_single_hex_uniaxial.py``
    * - Abaqus Verification Manual
      - ``5.00 × 10⁻⁶``
      - AVM 1.3.1 (uniaxial stress, C3D8)
    * - MAPDL Verification Manual
      - ``5.00 × 10⁻⁶``
      - VM-1 (statically indeterminate reaction force analysis)

Every source agrees to the precision at which the value is
stated.  This is the minimum bar an FE implementation has to
clear — no solver that ships results differing from Hooke's law
on a single element is worth trusting.

Source
------

Backing regression test:
:file:`tests/analytical/test_single_hex_uniaxial.py`
— landed with Agent 2's TA-9b analytical suite (#150).