For this kind of calculation, MAC LAM may calculate for a given multi-layered material:

• The ABBD matrix and its inverse. The generalized constitutive equation of the multi-layered material involves this matrix:

(1)

where , and are 3 X 3 matrixes: N_x, N_y and N_xy are the membranar forces; M_x, M_y and M_xy are the in-plane generalized moments, e_x, e_y and e_xy are the generalized in-plane strains; c_x, c_y and c_xy are the curvatures.

• The out-of-plane stiffness matrix and its inverse. This matrix appears in the generalized behavior of a multi-layered material

(2)

 

where Q_x, Q_y and Q_xy are the generalized shear forces, e_xz and e_yz are the generalized strains associated to the shear forces.

• Apparent Young's moduli in the global coordinates system (x, y, z)
• Apparent shear moduli in the global coordinates system (x, y, z)
• Apparent flexural moduli in the global coordinates system
• Apparent Poisson's ratio in the global coordinates system
• Apparent thermal expansion coefficients in the global coordinates system
• Apparent hygroscopic coefficients in the global coordinates system

Let's discuss in a more detailed way how to operate MAC LAM in order to obtain the previous results:

• First of all, the multi-layered material has to be defined
• then, the calculation and report options have to be selected
• after that, click on <<Analyze>> in the main window (see Figure 1)

Figure 1. Launching STIFFNESSES calculations.

• Then a window appears (see Figure 2) with the report containing the results and data selected in Calculation and report options.

Figure 2. Report of the STIFFNESSES calculation.

Calculation and report options

For the STIFFNESSES calculation type, there are options to which you may access by clicking on "Options" in the menu bar of the main window (see Figure 3).

Figure 3. Calculation and report options of STIFFNESSES calculation type.

Now the window on Figure 4 appears. Select everything you want to have in the report and the way you want to calculate the out-of-plane stiffness matrix . This matrix may be calculated by the inverse of the sum of the compliances of the layers (as advised in [2] ) or by the layer stiffnesses sum.

Figure 4. Selecting the calculation and report options for the STIFFNESSES calculation type.

Among the options you will find

• Show S and Sbar matrixes for each layer. S is the matrix of in-plane compliances calculated in the local coordinates system, L, T, N, (these axes are the othotropy directions). Sbar is the matrix of in-plane compliances calculated in the global coordinates system (x, y, z).
• Show Q and Qbar matrixes for each layer. Q is the inverse of the S matrix calculated in the local coordinates system, L, T, N. Qbar matrix is the inverse of the Sbar matrix calculated in the global coordinates system (x, y, z).
• Other notations whose meaning is evident.

References

[1] R.M. Jones, "Mechanics of composite materials", second edition, Taylor & Francis, United States, 1999.

[2] JF. Caron, K. Sab. Un nouveau modèle de plaque multicouche épaisse. Comptes Rendus de l'Académie des Sciences Paris II b, 329, 595-600, 2001.