II.1.5 In-plane and flexural laminate behavior

The classical laminate analysis is based on the assumption that in-plane and flexural behavior of the laminate is not related to out-of-plane shear loading. The corresponding laminate properties can be studied separately. The same remark is true for the load response calculation. In this section, the in-plane and flexural behavior of laminates are studied.

In this section the thermal and moisture expansions are not taken into account. The out-of-plane shear properties and loading of laminates is also discussed in a separate section. The results of classical laminate analysis are summarized here. The reader shall refer to the literature if more information on the developments that led to these results is needed. In this section, the different equations are written in laminate axes and the corresponding indices are noted $x$ and $y$.

Laminate compliance and stiffness matrices relate the in-plane forces and bending moments on one hand to the average strain and curvatures on the other hand. Those different quantities are defined as follows:

• In-plane normal forces tensor:

  $${\left\{N\right\}}_{\text{lam}}=\left(\begin{array}{c}\hfill N_{xx}\hfill \\ \hfill N_{yy}\hfill \\ \hfill N_{xy}\hfill \end{array}\right)={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}\left(\begin{array}{c}\hfill {\sigma }_{xx}\hfill \\ \hfill {\sigma }_{yy}\hfill \\ \hfill {\tau }_{xy}\hfill \end{array}\right)\text{d}z.$$

• Bending moment tensor:

  $${\left\{M\right\}}_{\text{lam}}=\left(\begin{array}{c}\hfill M_{xx}\hfill \\ \hfill M_{yy}\hfill \\ \hfill M_{xy}\hfill \end{array}\right)={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z\left(\begin{array}{c}\hfill {\sigma }_{xx}\hfill \\ \hfill {\sigma }_{yy}\hfill \\ \hfill {\tau }_{xy}\hfill \end{array}\right)\text{d}z.$$

  (II.1.31)

• Average deformation tensor:

  $${\left\{{ϵ}^0\right\}}_{\text{lam}}=\left(\begin{array}{c}\hfill {ϵ}_{xx}^0\hfill \\ \hfill {ϵ}_{yy}^0\hfill \\ \hfill {\gamma }_{xy}^0\hfill \end{array}\right)=\frac{1}{h}{\int \; <!--nolimits-->}_{-h∕2}^{h∕2}\left(\begin{array}{c}\hfill {ϵ}_{xx}\hfill \\ \hfill {ϵ}_{yy}\hfill \\ \hfill {\gamma }_{xy}\hfill \end{array}\right)\text{d}z=\left(\begin{array}{c}\hfill u_{,x}^0\hfill \\ \hfill v_{,y}^0\hfill \\ \hfill u_{,y}^0+v_{,x}^0\hfill \end{array}\right).$$

• Curvature tensor:

  $${\left\{\kappa \right\}}_{\text{lam}}=\left(\begin{array}{c}\hfill {\kappa }_{xx}\hfill \\ \hfill {\kappa }_{yy}\hfill \\ \hfill {\kappa }_{xy}\hfill \end{array}\right)=\frac{12}{h^3}{\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z\left(\begin{array}{c}\hfill {ϵ}_{xx}\hfill \\ \hfill {ϵ}_{yy}\hfill \\ \hfill {\gamma }_{xy}\hfill \end{array}\right)\text{d}z=\left(\begin{array}{c}\hfill w_{,xx}^0\hfill \\ \hfill w_{,yy}^0\hfill \\ \hfill 2w_{,xy}^0\hfill \end{array}\right).$$

  (II.1.32)

Note that the average strain tensor, as well as the true tensor, are not “real” tensors because their shear components (i.e. the non-diagonal components) are angular components.

The relations between the four tensors are then given by two equations:

The different matrices and vectors introduced in these equations are defined below:

• Matrix ${\left[A\right]}_{\text{lam}}$ is a $3\times 3$ matrix corresponding to the in-plane stiffness of laminate. Its components are calculated as follows:

  $${\left[A\right]}_{\text{lam}}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}{\left[C\right]}_{\text{lam}}(z)\text{d}z=\sum _{k=1}^N{\left[C_k\right]}_{\text{lam}}{e}_k.$$

• Matrix ${\left[D\right]}_{\text{lam}}$ is a $3\times 3$ matrix corresponding to the flexural stiffness of laminate. Its components are calculated as follows:

  $${\left[D\right]}_{\text{lam}}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}{z}^2{\left[C\right]}_{\text{lam}}(z)\text{d}z=\sum _{k=1}^N{\left[C_k\right]}_{\text{lam}}\left(\frac{z_k^3-z_{k-1}^3}{3}\right).$$

• Matrix ${\left[B\right]}_{\text{lam}}$ is a $3\times 3$ matrix corresponding to the coupling between flexural and in-plane behavior of the laminate. It is calculated as follows:

  $${\left[B\right]}_{\text{lam}}={\int \; <!--nolimits-->}_{-h∕2}^{h∕2}z{\left[C\right]}_{\text{lam}}(z)\text{d}z=\sum _{k=1}^N{\left[C_k\right]}_{\text{lam}}\left(\frac{z_k^2-z_{k-1}^2}{2}\right).$$

All the new matrices and vectors are obtained by summation of the ply contributions. In order to obtain the ply stiffness matrix in laminate axes ${\left[C\right]}_{\text{lam}}^k$ and the ply thermo-elastic CTE coefficients in thermo-elastic axes ${\left\{\alpha \right\}}_{\text{lam}}^k$, the transformations (II.1.24) and (II.1.26) are used respectively. Note however that if a ply is characterized by an orientation $\theta$ with respect to the laminate axes, the rotation of ply properties must be of an angle $-\theta$.

The laminate compliance matrices ${\left[a\right]}_{\text{lam}}$, ${\left[b\right]}_{\text{lam}}$ and ${\left[d\right]}_{\text{lam}}$ are obtained by inversion of the $6\times 6$ ${\left[ABBD\right]}_{\text{lam}}$ matrix:

Then, the average laminate strain and its curvature tensor can be calculated as follows:

Equivalent moduli corresponding to the calculated stiffness matrices $A$ and $D$ are often computed. We follow the expressions presented in [Pal99]:

• The normalized in-plane, coupling and flexural stiffness and compliance matrices are calculated as follows:

  $${\left[A^{*}\right]}_{\text{lam}}=\frac{1}{h}{\left[A\right]}_{\text{lam}},{\left[a^{*}\right]}_{\text{lam}}=h{\left[a\right]}_{\text{lam}},$$

  $${\left[B^{*}\right]}_{\text{lam}}=\frac{2}{h^2}{\left[B\right]}_{\text{lam}},{\left[b^{*}\right]}_{\text{lam}}=\frac{h^2}{2}{\left[b\right]}_{\text{lam}},$$

  $${\left[D^{*}\right]}_{\text{lam}}=\frac{12}{h^3}{\left[D\right]}_{\text{lam}},{\left[d^{*}\right]}_{\text{lam}}=\frac{h^3}{12}{\left[d\right]}_{\text{lam}}.$$

• Equivalent in-plane moduli and Poisson ratios are then given by:

  $$E_{xx}=\frac{1}{a_{11}^{*}},E_{yy}=\frac{1}{a_{22}^{*}},G_{xy}=\frac{1}{a_{33}^{*}},$$

  $${\nu }_{xy}=-\frac{a_{33}^{*}}{a_{11}^{*}},{\nu }_{yx}=-\frac{a_{33}^{*}}{a_{22}^{*}}.$$

  These moduli correspond to a case for which the laminate is free to bend under in-plane loading. This can be the case when there is coupling of in-plane and flexural laminate behavior. (Matrix ${\left[B\right]}_{\text{lam}}$ is not zero.) The laminate in-plane engineering constants with suppressed curvature are calculated as follows:

  $${\left[{\tilde{a}}^{*}\right]}_{\text{lam}}={\left[A^{*}\right]}_{\text{lam}}^{-1},$$

  $$E_{xx}^{\kappa =0}=\frac{1}{{\tilde{a}}_{11}^{*}},E_{yy}^{\kappa =0}=\frac{1}{{\tilde{a}}_{22}^{*}},G_{xy}^{\kappa =0}=\frac{1}{{\tilde{a}}_{33}^{*}},$$

  $${\nu }_{xy}^{\kappa =0}=-\frac{{\tilde{a}}_{33}^{*}}{{\tilde{a}}_{11}^{*}},{\nu }_{yx}^{\kappa =0}=-\frac{{\tilde{a}}_{33}^{*}}{{\tilde{a}}_{22}^{*}}.$$

• Similarly, equivalent flexural moduli and Poisson ratios can be calculated. Note the following relation:

  $${\left({\left[D\right]}_{\text{lam}}\right)}^{-1}=\left(\begin{array}{ccc}\hfill 1∕{\overline{EI}}_{11}\hfill & \hfill 1∕{\overline{EI}}_{12}\hfill & \hfill 1∕{\overline{EI}}_{13}\hfill \\ \hfill 1∕{\overline{EI}}_{21}\hfill & \hfill 1∕{\overline{EI}}_{22}\hfill & \hfill 1∕{\overline{EI}}_{23}\hfill \\ \hfill 1∕{\overline{EI}}_{31}\hfill & \hfill 1∕{\overline{EI}}_{32}\hfill & \hfill 1∕{\overline{EI}}_{33}\hfill \end{array}\right).$$

  If there is no coupling between laminate flexural and membrane properties

  $${\left({\left[D\right]}_{\text{lam}}\right)}^{-1}={\left[d\right]}_{\text{lam}}=\left(\begin{array}{ccc}\hfill 1∕{\overline{EI}}_{11}\hfill & \hfill 1∕{\overline{EI}}_{12}\hfill & \hfill 1∕{\overline{EI}}_{13}\hfill \\ \hfill 1∕{\overline{EI}}_{21}\hfill & \hfill 1∕{\overline{EI}}_{22}\hfill & \hfill 1∕{\overline{EI}}_{23}\hfill \\ \hfill 1∕{\overline{EI}}_{31}\hfill & \hfill 1∕{\overline{EI}}_{32}\hfill & \hfill 1∕{\overline{EI}}_{33}\hfill \end{array}\right).$$

  Generally, the “no-coupling” behavior is assumed. (See for example section II.1.6.) Therefore, one can simply write:

  $$E_{xx}^f=\frac{1}{d_{11}^{*}},E_{yy}^f=\frac{1}{d_{22}^{*}},G_{xy}^f=\frac{1}{d_{33}^{*}},$$

  $${\nu }_{xy}^f=-\frac{d_{33}^{*}}{d_{11}^{*}},{\nu }_{yx}^f=-\frac{d_{33}^{*}}{d_{22}^{*}}.$$