Defining a laminate: conventions
================================

A laminate is usually created with :func:`.laminated_plate`, from the ply
angles, thicknesses, material properties and densities, or with
:func:`.isotropic_plate` for a single isotropic layer::

    from composites import laminated_plate

    laminaprop = (142e9, 8.7e9, 0.28, 5.1e9, 5.1e9, 3.2e9)
    lam = laminated_plate([0, 45, -45, 90], plyt=0.125e-3,
                          laminaprop=laminaprop, rho=1600., offset=0.)

The conventions below apply to every :class:`.Laminate`.


Stacking sequence and thickness coordinate
------------------------------------------

The angles of ``stack``, and the lists ``plyts``, ``laminaprops`` and
``rhos``, run from the bottom to the top ply. The coordinate `z` is normal to
the reference surface, pointing from the bottom to the top face, and it is
measured from the reference surface, which is the surface of the
generalized strains, see :doc:`constitutive_matrices`.


Offset
------

``offset`` is the position of the mid-surface relative to the reference
surface, `d` below. The plies run from

.. math::

    z_1 = -\frac{h}{2} + d \quad \text{to} \quad z_{N+1} = +\frac{h}{2} + d

such that ``offset = 0`` puts the reference surface at the mid-surface,
``offset = +h/2`` at the bottom face and ``offset = -h/2`` at the top face.
For example, a skin modelled on its mould line, with the laminate above it,
has ``offset = h/2``. With an offset, ``B`` is not zero even for an
isotropic plate, `B = d h C`, see :doc:`ex_isotropic_offset`, and the same
holds for the odd moments of the density, see :doc:`mass_integrals`.
:meth:`.Laminate.make_balanced`, :meth:`.Laminate.make_orthotropic`,
:meth:`.Laminate.make_symmetric` and :meth:`.Laminate.make_smeared` require
``offset = 0``.


Material properties
-------------------

``laminaprop`` is read by :func:`.read_laminaprop`, with 2, 6 or 9 entries:

- ``(E, nu)``: isotropic, with ``G = E/(2*(1 + nu))`` for the three shear
  moduli;
- ``(E11, E22, nu12, G12, G13, G23)``: orthotropic ply in plane stress, the
  usual case;
- ``(E11, E22, nu12, G12, G13, G23, E33, nu13, nu23)``: orthotropic ply with
  the properties of the three-dimensional constitutive matrix of
  :meth:`.MatLamina.get_constitutive_matrix`.

The index 1 is the fibre direction, 2 the in-plane transverse direction and 3
the normal direction. The ply stiffnesses used in the laminate integrals are
always the plane-stress ones, see :meth:`.Lamina.rebuild`; ``G13`` and
``G23`` give the transverse shear stiffnesses ``q44L``, ``q45L``, ``q55L``.


Ply angle
---------

The ply angle `\theta`, in degrees, is measured from the laminate `x` axis to
the material direction 1, positive from `x` towards `y`, i.e. a positive
rotation about `z`: the direction 1 is `\{\cos\theta, \sin\theta\}` in the
laminate axes. Reversing the sign of all the angles changes the sign of the
16 and 26 terms and of ``A45``.


Units
-----

No units are assumed, and any consistent system can be used, e.g. N, m and
kg, with the densities in kg/m\ :sup:`3`, or N, mm and t, with the densities
in t/mm\ :sup:`3`. The dimension of each attribute is given in the
documentation of :class:`.Laminate`.


Ordering of stresses and strains
--------------------------------

- In-plane: `\{\sigma_{xx}, \sigma_{yy}, \tau_{xy}\}` and `\{\varepsilon_{xx},
  \varepsilon_{yy}, \gamma_{xy}\}`, with the engineering shear strain
  `\gamma_{xy} = 2 \varepsilon_{xy}`.
- Transverse shear: `\{\tau_{yz}, \tau_{xz}\}`, `\{\gamma_{yz},
  \gamma_{xz}\}`, with `\gamma = 2 \varepsilon`, and `\{Q_y, Q_x\}`; the
  index 4 refers to `yz` and the index 5 to `xz`.
- :attr:`.Laminate.ABD` relates `\{N_{xx}, N_{yy}, N_{xy}, M_{xx}, M_{yy},
  M_{xy}\}` to `\{\varepsilon^{(0)}_{xx}, \varepsilon^{(0)}_{yy},
  \gamma^{(0)}_{xy}, \varepsilon^{(1)}_{xx}, \varepsilon^{(1)}_{yy},
  \gamma^{(1)}_{xy}\}`, the same order used by the strain gradients of
  :meth:`.Laminate.calc_equilibrium_transverse_shear`.
- :meth:`.Lamina.get_constitutive_matrix` is ``5 x 5``, in the laminate axes,
  ordered as `\{xx, yy, xy, yz, xz\}`.
- :meth:`.MatLamina.get_constitutive_matrix` is ``6 x 6``, in the material
  axes, ordered as `\{11, 22, 33, 23, 13, 12\}`.
- :meth:`.Laminate.calc_transverse_shear_stress` receives ``Qy, Qx`` and
  returns ``tau_yz, tau_xz``.