Laminate constitutive matrices per plate theory

A Laminate contains the thickness integrals of the ply stiffnesses required by three equivalent single-layer theories: the classical laminated plate theory (CLPT), the first-order shear deformation theory (FSDT) and the third-order shear deformation theory (TSDT) of Reddy [1]. All of them are computed at once by Laminate.calc_constitutive_matrix(), which is called by laminated_plate() and isotropic_plate(), such that the same object serves a model of any of the three theories.

Displacement fields and generalized strains

With \(z\) measured from the reference surface, the in-plane displacements of the three theories are [1]:

\[\begin{split}\text{CLPT:} \quad u = u_0 - z w_{,x}, \quad v = v_0 - z w_{,y} \\ \text{FSDT:} \quad u = u_0 + z \phi_x, \quad v = v_0 + z \phi_y \\ \text{TSDT:} \quad u = u_0 + z \phi_x - \frac{4}{3h^2} z^3 (\phi_x + w_{,x}), \quad v = v_0 + z \phi_y - \frac{4}{3h^2} z^3 (\phi_y + w_{,y})\end{split}\]

and \(w(x, y)\) in all of them. The in-plane strains \(\varepsilon = \{\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy}\}\) and the transverse shear strains \(\gamma = \{\gamma_{yz}, \gamma_{xz}\}\) are written with the generalized strains of the reference surface as:

\[\varepsilon = \varepsilon^{(0)} + z \varepsilon^{(1)} + z^3 \varepsilon^{(3)}, \qquad \gamma = \gamma^{(0)} + z^2 \gamma^{(2)}\]

where \(\varepsilon^{(3)}\) and \(\gamma^{(2)}\) exist only in the TSDT, and \(\gamma^{(0)}\) vanishes in the CLPT. The FSDT gives a constant \(\gamma^{(0)}\) through the thickness, which requires a shear correction, see Choosing the transverse shear correction, whereas the TSDT satisfies the traction-free conditions at \(z = \pm h/2\) and needs no shear correction.

Thickness integrals

With \(C^{(k)}\) the plane-stress stiffness of ply \(k\) in the laminate axes and \(C_s^{(k)}\) its transverse shear stiffness, both given by Lamina.rebuild(), and the ply occupying \(z_k \le z \le z_{k+1}\):

\[\begin{split}(A, B, D, E, F, H) = \sum_k \int_{z_k}^{z_{k+1}} C^{(k)} (1, z, z^2, z^3, z^4, z^6) \, dz \\ (\bar{A}_{ts}, D_{ts}, F_{ts}) = \sum_k \int_{z_k}^{z_{k+1}} C_s^{(k)} (1, z^2, z^4) \, dz\end{split}\]

The FSDT does not use \(\bar{A}_{ts}\) directly but the corrected \(A_{ts}\), see Laminate.calc_transverse_shear_stiffness().

Constitutive relations

The stress resultants \(N\), \(M\), \(P\) are the integrals of \(\{\sigma_{xx}, \sigma_{yy}, \tau_{xy}\}\) times \(1\), \(z\) and \(z^3\), and \(Q\), \(R\) those of \(\{\tau_{yz}, \tau_{xz}\}\) times \(1\) and \(z^2\), such that \(Q = \{Q_y, Q_x\}\). The strain energy per unit area of the reference surface is half the product of the resultants by the generalized strains, and the constitutive relations become:

\[\begin{split}\text{CLPT:} \quad \begin{Bmatrix} N \\ M \end{Bmatrix} = \begin{bmatrix} A & B \\ B & D \end{bmatrix} \begin{Bmatrix} \varepsilon^{(0)} \\ \varepsilon^{(1)} \end{Bmatrix}\end{split}\]

\[\begin{split}\text{FSDT:} \quad \begin{Bmatrix} N \\ M \end{Bmatrix} = \begin{bmatrix} A & B \\ B & D \end{bmatrix} \begin{Bmatrix} \varepsilon^{(0)} \\ \varepsilon^{(1)} \end{Bmatrix}, \qquad Q = A_{ts} \gamma^{(0)}\end{split}\]

\[\begin{split}\text{TSDT:} \quad \begin{Bmatrix} N \\ M \\ P \end{Bmatrix} = \begin{bmatrix} A & B & E \\ B & D & F \\ E & F & H \end{bmatrix} \begin{Bmatrix} \varepsilon^{(0)} \\ \varepsilon^{(1)} \\ \varepsilon^{(3)} \end{Bmatrix}, \qquad \begin{Bmatrix} Q \\ R \end{Bmatrix} = \begin{bmatrix} \bar{A}_{ts} & D_{ts} \\ D_{ts} & F_{ts} \end{bmatrix} \begin{Bmatrix} \gamma^{(0)} \\ \gamma^{(2)} \end{Bmatrix}\end{split}\]

The transverse shear matrices are ordered as [[44, 45], [45, 55]], with the index 4 corresponding to \(yz\) and the index 5 to \(xz\), in the same order as \(\gamma = \{\gamma_{yz}, \gamma_{xz}\}\) and \(Q = \{Q_y, Q_x\}\). Some references, e.g. Rohwer [2], use the opposite order \(\{xz, yz\}\).

Symbols and attributes

Symbol	Laminate attribute	Theories
\(A\), \(B\), \(D\)	Laminate.A, Laminate.B, Laminate.D, A11 to D66	CLPT, FSDT, TSDT
\([[A, B], [B, D]]\)	Laminate.ABD	CLPT, FSDT
\(E\), \(F\), \(H\)	Laminate.E, Laminate.F, Laminate.H, E11 to H66	TSDT
\(A_{ts}\), corrected	Laminate.Ats, A44, A45, A55	FSDT
\(\bar{A}_{ts}\), uncorrected	Laminate.Abar_ts, Abar44, Abar45, Abar55	TSDT
\(D_{ts}\)	Laminate.Dts, D44, D45, D55	TSDT
\(F_{ts}\)	Laminate.Fts, F44, F45, F55	TSDT
\(\bar{\bar{A}}_{ts}\), constant stress	Laminate.Abarbar_ts	for comparison
\(C^{(k)}\)	q11L, q12L, q16L, q22L, q26L, q66L of Lamina	all
\(C_s^{(k)}\)	q44L, q45L, q55L of Lamina	all
\(h\), offset of the mid-surface	h, offset	all

The subscripts 16, 26 and 66 refer to the \(xy\) components. In the transverse shear blocks of the TSDT, some references denote \(\bar{A}_{ts}\), \(D_{ts}\) and \(F_{ts}\) with the same letters \(A\), \(D\) and \(F\) used for the in-plane stiffnesses; in composites they have their own names. The moments of the density used in the mass matrices are described in Thickness integrals for mass matrices.

Assembling the matrices in a model

A finite element or Ritz model gathers the generalized strains in one vector \(e\) and the blocks above in one symmetric matrix \(K_c\), such that the strain energy per unit area is \(e^T K_c e / 2\) and the stiffness matrix is \(\int B^T K_c B \, dA\), where \(B\) relates the degrees of freedom to \(e\). The common layouts are:

• CLPT, 6 x 6: \(e = \{\varepsilon^{(0)}, \varepsilon^{(1)}\}\), \(K_c =\) lam.ABD.

• FSDT, 8 x 8: \(e = \{\varepsilon^{(0)}, \varepsilon^{(1)}, \gamma^{(0)}\}\), K_c = [[ABD, 0], [0, Ats]].

• TSDT, 13 x 13: \(e = \{\varepsilon^{(0)}, \varepsilon^{(1)}, \varepsilon^{(3)}, \gamma^{(0)}, \gamma^{(2)}\}\), K_c = [[A, B, E, 0, 0], [B, D, F, 0, 0], [E, F, H, 0, 0], [0, 0, 0, Abar_ts, Dts], [0, 0, 0, Dts, Fts]].

These are, for instance, the layouts of the Ritz models of panels. In an FSDT model, a shear correction factor of the user’s choice is applied to Abar_ts instead of Ats, see Choosing the transverse shear correction. The assembly of the TSDT matrix is shown in Matrices of the third-order shear deformation theory.

The TSDT displacement field above, with the traction-free conditions at \(z = \pm h/2\), takes the reference surface at the mid-surface, whereas all the matrices of a Laminate are integrated with \(z\) measured from the reference surface, i.e. including offset, see Defining a laminate: conventions.

References

[1] (1,2)

Reddy, J. N., Mechanics of Laminated Composite Plates and Shells, CRC Press, 2003. https://doi.org/10.1201/b12409

[2]

Rohwer, K., Improved Transverse Shear Stiffnesses for Layered Finite Elements, DFVLR-FB 88-32, 1988.