Isotropic plate with an offset

For a homogeneous isotropic plate of thickness \(h\) whose mid-surface is at \(z = d\) from the reference surface, i.e. offset = d, the thickness integrals have the closed form

\[A = h C, \qquad B = d h C, \qquad D = h \left( d^2 + \frac{h^2}{12} \right) C, \qquad A_{ts} = \frac{5}{6} G h I\]

where \(C\) is the plane-stress stiffness of the material and \(I\) the 2x2 identity matrix. The offset creates a membrane-bending coupling \(B \neq 0\) and, in the same way, the odd moments of the density, see Thickness integrals for mass matrices. The test below checks these expressions, the matrices \(E\), \(F\), \(H\) of the TSDT and the six mass integrals against the integrals \(\int_{d - h/2}^{d + h/2} z^n dz\):

def test_example_isotropic_plate_offset():
    # closed form of an isotropic plate whose mid-surface is at z = d
    E, nu, h, d, rho = 70e9, 0.3, 2e-3, 0.6e-3, 2700.
    lam = isotropic_plate(thickness=h, E=E, nu=nu, offset=d, rho=rho)
    C = E/(1 - nu**2)*np.array([[1, nu, 0], [nu, 1, 0], [0, 0, (1 - nu)/2]])
    G = E/(2*(1 + nu))
    atol = 1e-12*E
    assert np.allclose(lam.A, h*C, rtol=1e-12, atol=atol*h)
    assert np.allclose(lam.B, d*h*C, rtol=1e-12, atol=atol*h*d)
    assert np.allclose(lam.D, h*(d**2 + h**2/12)*C, rtol=1e-12, atol=atol*h**3)
    # the shear correction of an isotropic plate is 5/6
    assert np.allclose(lam.Ats, 5/6*G*h*np.eye(2), rtol=1e-12, atol=atol*h)
    # with s uniform in [-h/2, h/2]: <s^2> = h^2/12, <s^4> = h^4/80,
    # <s^6> = h^6/448, and int (d + s)^n ds = h <(d + s)^n>
    m = [1., d, d**2 + h**2/12, d**3 + d*h**2/4,
         d**4 + d**2*h**2/2 + h**4/80, None,
         d**6 + 15/12*d**4*h**2 + 15/80*d**2*h**4 + h**6/448]
    assert np.allclose(lam.E, h*m[3]*C, rtol=1e-12, atol=atol*h**4)
    assert np.allclose(lam.F, h*m[4]*C, rtol=1e-12, atol=atol*h**5)
    assert np.allclose(lam.H, h*m[6]*C, rtol=1e-12, atol=atol*h**7)
    for name, n in (('intrho', 0), ('intrhoz', 1), ('intrhoz2', 2),
                    ('intrhoz3', 3), ('intrhoz4', 4), ('intrhoz6', 6)):
        assert np.isclose(getattr(lam, name), rho*h*m[n], rtol=1e-12, atol=0)
    # without offset the odd moments vanish
    lam0 = isotropic_plate(thickness=h, E=E, nu=nu, rho=rho)
    assert abs(lam0.intrhoz) < 1e-15*rho*h**2
    assert abs(lam0.intrhoz3) < 1e-15*rho*h**4
    assert np.allclose(lam0.B, 0, atol=atol*h**2)