Thick-walled cylinder under internal pressure (Lamé)
======================================================

One of the oldest pressure-vessel benchmarks in solid mechanics.
A long cylinder with inner radius :math:`a`, outer radius
:math:`b`, subjected to internal pressure :math:`p_i` (no
external pressure) develops radial and hoop stress fields that
Lamé derived analytically in 1852.

Under the plane-strain assumption (``ε_z = 0`` — long cylinder
with ends restrained):

.. math::

    \sigma_r(r) &= \frac{p_i a^{2}}{b^{2} - a^{2}}
                    \Bigl(1 - \frac{b^{2}}{r^{2}}\Bigr), \\
    \sigma_\theta(r) &= \frac{p_i a^{2}}{b^{2} - a^{2}}
                    \Bigl(1 + \frac{b^{2}}{r^{2}}\Bigr), \\
    u_r(r) &= \frac{p_i\,a^{2}\,r}{E(b^{2}-a^{2})}
              \Bigl[(1 - \nu - 2\nu^{2})
              + \frac{b^{2}(1+\nu)}{r^{2}}\Bigr].

Radial displacement peaks at the inner surface; hoop stress also
peaks there.  :math:`\sigma_r(a) = -p_i` exactly (the applied
pressure, with compression sign convention).

Problem
-------

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

    * - Parameter
      - Value
    * - Inner radius ``a``
      - 0.01 m (10 mm)
    * - Outer radius ``b``
      - 0.02 m (20 mm)
    * - Axial slab thickness
      - 0.001 m (plane-strain model)
    * - Young's modulus ``E``
      - 210 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Internal pressure ``p_i``
      - 100 MPa
    * - Expected ``u_r(a)``
      - 9.08 × 10⁻⁶ m
    * - Expected ``σ_θ(a)``
      - 166.7 MPa (tensile)
    * - Expected ``σ_θ(b)``
      - 66.7 MPa (tensile)

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

Ran by :file:`tests/validation/test_lame_cylinder.py` on a
mapped quarter-annulus hex mesh (``SOLID185`` kernel).  Plane
strain is enforced by pinning ``UZ = 0`` on every node of the
thin slab; symmetry BCs on the cut x-axis / y-axis planes
restore the full annulus.

The internal pressure is lumped onto the inner-surface nodes via
trapezoid arc-length weighting, with local nodal accumulation
before calling ``Model.apply_force`` (the ``_f_impl`` path
assigns rather than accumulates).

.. list-table::
    :header-rows: 1
    :widths: 15 15 15 20 15 20

    * - Refinement
      - Mesh (n_θ × n_r)
      - ``u_r(a)``
      - Δ
      - ``σ_θ(a)``
      - Δ
    * - Coarse
      - 16 × 4
      - 9.038 × 10⁻⁶ m
      - −0.45 %
      - 186.0 MPa
      - +11.58 %
    * - Medium
      - 32 × 8
      - 9.069 × 10⁻⁶ m
      - −0.12 %
      - 177.6 MPa
      - +6.57 %
    * - Reference
      - 64 × 16
      - 9.077 × 10⁻⁶ m
      - −0.03 %
      - 172.5 MPa
      - +3.51 %

Radial displacement converges to under 0.05 %.  The hoop-stress
concentration at ``r = a`` is harder to resolve on a coarse mesh
— the ``1/r²`` gradient is steep near the inner surface — but
refines to ~3.5 % at the reference mesh, well inside the 8 %
engineering tolerance.

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

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

    * - Source
      - Reported ``σ_θ(a)`` (MPa)
      - Problem ID / location
    * - Closed form (Lamé 1852)
      - 166.67
      - Timoshenko & Goodier 1970 §33
    * - Timoshenko & Goodier (1970)
      - 166.67
      - *Theory of Elasticity* §33
    * - femorph-solver (refined)
      - 172.5
      - :file:`test_lame_cylinder.py`
    * - MAPDL Verification Manual
      - ≈ 166.7
      - VM-25 *Stresses in a long cylinder under internal pressure*
    * - Abaqus Verification Manual
      - ≈ 166.7
      - AVM 1.1.14 thick-walled cylinder family

Source
------

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

Backing regression test:
:file:`tests/validation/test_lame_cylinder.py` — convergence
study across three mesh refinements asserting all three
published quantities (``u_r(a)``, ``σ_θ(a)``, ``σ_θ(b)``)
pass their tolerances and that ``u_r`` converges monotonically.