Thickness integrals for mass matrices#

The consistent mass matrix of an equivalent single-layer model follows from the kinetic energy

\[T = \frac{1}{2} \int_V \rho \left( \dot{u}^2 + \dot{v}^2 + \dot{w}^2 \right) dV\]

where the displacements \(u\), \(v\), \(w\) are those of the theory, see Laminate constitutive matrices per plate theory. Integrating through the thickness separately from the integration over the reference surface, the only laminate data left are the moments of the density

\[I_n = \sum_k \int_{z_k}^{z_{k+1}} \rho^{(k)} z^n \, dz\]

computed by Laminate.calc_constitutive_matrix() with \(z\) measured from the reference surface, as the stiffnesses:

\(n\)

attribute

dimension

0

intrho

mass per unit area

1

intrhoz

mass per unit area x length

2

intrhoz2

mass per unit area x length^2

3

intrhoz3

mass per unit area x length^3

4

intrhoz4

mass per unit area x length^4

6

intrhoz6

mass per unit area x length^6

Moments required by each theory#

In the FSDT, \(u = u_0 + z \phi_x\), and through the thickness:

\[\int \rho \dot{u}^2 dz = I_0 \dot{u}_0^2 + 2 I_1 \dot{u}_0 \dot{\phi}_x + I_2 \dot{\phi}_x^2, \qquad \int \rho \dot{w}^2 dz = I_0 \dot{w}^2\]

and the same with \(v_0\) and \(\phi_y\). The CLPT follows with \(\phi_x = -w_{,x}\) and \(\phi_y = -w_{,y}\). Both theories require \(I_0\), \(I_1\) and \(I_2\), i.e. intrho, intrhoz and intrhoz2.

In the TSDT, \(u = u_0 + z \phi_x + c z^3 \psi_x\), with \(c = -4/(3h^2)\) and \(\psi_x = \phi_x + w_{,x}\):

\[\int \rho \dot{u}^2 dz = I_0 \dot{u}_0^2 + 2 I_1 \dot{u}_0 \dot{\phi}_x + I_2 \dot{\phi}_x^2 + 2 c I_3 \dot{u}_0 \dot{\psi}_x + 2 c I_4 \dot{\phi}_x \dot{\psi}_x + c^2 I_6 \dot{\psi}_x^2\]

and the same in \(y\), such that \(I_0\), \(I_1\), \(I_2\), \(I_3\), \(I_4\) and \(I_6\) are required, all the attributes of the table above.

In shells, the rotations that enter the in-plane displacements in \(T\) must be the same as those used in the strains, e.g. including the corrections of Sanders’ kinematics in a cylinder, such that a rigid-body motion gives the exact velocity of every point of the thickness.

Effect of the offset#

For a density symmetric about the reference surface the odd moments \(I_1\) and \(I_3\) vanish. With offset different than zero, or an unsymmetric distribution of the ply densities, \(I_1 \neq 0\) couples the in-plane and the rotary inertias in the same way as B couples the membrane and bending stiffnesses. For a homogeneous plate whose mid-surface is at \(z = d\):

\[I_0 = \rho h, \qquad I_1 = \rho h d, \qquad I_2 = \rho h \left( d^2 + \frac{h^2}{12} \right)\]

see Isotropic plate with an offset, where the higher moments are also checked.

Densities of the plies#

The density of each ply is the rho of its MatLamina. With laminated_plate(), rho gives the same density to all the plies and rhos one density per ply, e.g. for a sandwich with a light core. The default density is zero, giving zero mass integrals, and the shear correction 'thickness_shear' requires positive densities, see Choosing the transverse shear correction.

The mass integrals are computed only from the plies: a Laminate created from lamination parameters, see laminate_from_LaminationParameters(), or by double_double_laminate() or composites.utils.n_double_laminate() has no plies, and its mass integrals are zero.