Plate with a circular hole under uniaxial tension — Kirsch
=============================================================

Kirsch's 1898 analytical solution gives the canonical stress-
concentration factor ``K_t = 3`` for an infinite plate under
uniaxial tension ``σ_∞`` containing a circular hole.  The
stress peaks at the hole edge at the points perpendicular to
the applied tension (``θ = ±π/2``):

.. math::

    \sigma_{\theta\theta}^{\text{max}} = 3 \, \sigma_\infty.

In Cartesian coordinates, at the hole top ``(x=0, y=a)`` the
local tangential direction is ``-x``, so the peak stress shows
up as ``σ_xx = 3σ_∞``.  Traction-free hole edge: ``σ_rr = 0``,
which in the same basis is ``σ_yy = 0`` at the hole top.

The factor of 3 is the stress-concentration benchmark used
across every structural FEA verification manual.

Problem
-------

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

    * - Parameter
      - Value
    * - Hole radius ``a``
      - 0.1 m
    * - Plate half-width ``W``
      - 1.0 m (``a / W = 0.1``)
    * - Out-of-plane thickness
      - 0.01 m (plane stress)
    * - Young's modulus ``E``
      - 210 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Remote tension ``σ_∞``
      - 10 MPa (in +x)
    * - Expected ``σ_xx`` at hole top
      - 30 MPa (``K_t = 3``)

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

Ran by :file:`tests/validation/test_plate_with_hole.py` using
the ``QUAD4_PLANE`` plane-stress kernel on a quarter-symmetry
mesh.  Parametric grid: radial-graded mapping from the hole
boundary to the square outer boundary, with a geometric stretch
(``q = 1.25``) that concentrates nodes near the hole where the
stress gradient is steep.

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

    * - Refinement
      - ``n_θ × n_r``
      - ``σ_xx`` at hole top (MPa)
      - Δ vs ``K_t = 3 σ_∞``
    * - Coarse
      - 16 × 8
      - 30.70
      - +2.32 %
    * - Medium
      - 32 × 12
      - 31.50
      - +5.00 %
    * - Reference
      - 64 × 20
      - 30.98
      - +3.25 %

Converges to within ~3 % of ``K_t = 3 σ_∞`` at the reference
mesh — inside the 10 % engineering tolerance that accommodates
the finite-width correction at ``a / W = 0.1`` (Peterson gives
:math:`K_t \approx 3.04` for this ratio).

σ_xx is recovered from forward-difference strain estimates
on the parametric grid: ``ε_xx = u_x(θ_idx) / x(θ_idx)`` (y-
axis symmetry forces ``u_x(0, y) = 0``; the next θ-node has
``x > 0``) and ``ε_yy = (u_y(rad_idx) - u_y(hole_top)) / Δy``.
The framework's ``compute_nodal_stress`` helper doesn't yet
cover plane-element kernels (tracked as VERIFY-BLOCKED task
#177); this benchmark uses the local FD extraction pending
that fix.

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

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

    * - Source
      - Reported ``σ_xx`` at hole top (MPa)
      - Problem ID / location
    * - Closed form (Kirsch 1898)
      - 30.00
      - ``K_t = 3`` infinite plate
    * - Timoshenko & Goodier (1970)
      - 30.00
      - *Theory of Elasticity* §35
    * - Peterson (2008) — finite width
      - ≈ 30.4
      - ``K_t(a/W = 0.1) ≈ 3.04``
    * - femorph-solver (reference mesh)
      - 30.98
      - :file:`test_plate_with_hole.py`
    * - MAPDL Verification Manual
      - ≈ 30.0
      - VM-74 *Stress concentration around a hole*
    * - Abaqus Verification Manual
      - ≈ 30.0
      - AVM 1.3.6 plate-with-hole family
    * - NAFEMS
      - ≈ 30.0
      - NL2 plate-with-hole (linear-elastic limit)

Source
------

Problem class:
:class:`femorph_solver.validation.problems.PlateWithHole`.

Backing regression test:
:file:`tests/validation/test_plate_with_hole.py`.