Three application examples are shown below:

• Free edge problem
• Stress analysis in an adhesive joint
• Determination of the residual thermal interfacial stresses

Free edge problem

The free edge problem deals with a laminate subjected to a tensile load with free edges. In our case, it is a symmetrical multi-layered material (q₁,...,q_n)₅ made up of N = 2n layers of orthotropic materials. Keeping in mind the symmetries, the problem can be reduced to the determination of the generalized fields in the lower right fourth section of Figure 1.

Figure 1. Free edge problem of a symmetrical multi-layered material simplified by symmetry.

Let us take the example of a (0,90)_S made up of a carbon-epoxy UD composite material. The material properties in the orthotropy coordinates system are:

E_L = 150GPa, E_T = 10GPa, E_N = 10GPa
G_LT = 5GPa, G_LN = 5GPa, G_TN = 5GPa
n_LT = 0.3 y n_LN = 0.3

The layer thickness value is 1mm, the laminate width value is 10mm and the axial strain e = 1%.

These stages are to be followed for obtaining the generalized fields:

1. Definition of the multi-layered material. In main window of DEILAM, type 4 in the space for the total number of layers. Next, enter in the table the properties of the previously defined layers (see Figure 2).

Figure 2. Definition of a carbon-epoxy (0,90)_S laminate.

2. Definition of the calculation properties. Click on the DEILAM <<Option>> in the main window menu (see Figure 3). In this window, uncheck the 2 options in which thermal and hygroscopic strains are considered (see Figure 4). Select the option for all generalized displacements and stresses (see Figure 4).

Figure 3. Where to click so that the calculation properties can be defined?

Figure 4. Definition of the calculation properties.

3. Definition of loads. Click on the button <<Loads>> in the DEILAM main window (see Figure 5). From the list of predetermined configurations shown in the window, select the <<Tensile load with free edges>> option and type 1 in the space corresponding to the strain e (see Figure 6). Finally, click on the button <<Read the configurations>>.

Figure 5. Where to click for defining the loads?

Figure 6. Load configuration.

4. Definition of the meshing. Click on the <<Meshing of the width [0, b]>> button in the main window (see Figure 7). Select, from the window that appears, the <<Automatic>> option in the list of meshing types (see Figure 8). In the space for the value of width b (in our case, b equals the half of the laminate width as a result of the simplification by symmetry), enter 5 (see Figure 8).

Figure 7. Where to click for the definition of the meshing?

Figure 8. Definition of the meshing.

5. Execution of the calculations. Click on the <<Analyze>> button in the DEILAM main window (see Figure 9).

Figure 9. Where to click for executing the calculations?

6. Visualization of the results. After the previous stage, a new window appears. Click on the <<Fields to be displayed>> button in this window (see Figure 10), and, in the new window (see Figure 11), select from the <<Fields to be displayed>> list, the <<In-plane and out-of-plane forces per layer>> option. Check the box that corresponds to the Nⁱ_yy (membranar force: in-plane force) field, and enter 1-2 (for plotting the membranar forces in layers 1 and 2: layers at 0° and at 90°, respectively) in the corresponding space. Then click on the <<Draw>> button. You will see a graph as the one in Figure 12. To distinguish both curves, click on the <<Chart properties>> button and select the colors and types of line you wish in order to make a difference between the curves.

Figure 10. Where to click for selecting the fields to be plotted in the chart?

Figure 11. Configuration for plotting the graph.

Figure 12. Plotting the curves of the selected fields.

7. Visualization of other fields. To visualize the plots of other fields such as interfacial stresses or generalized displacements, just retake stage 6 and select the desired fields.

Analysis of stresses in an adhesive joint

Let us consider the following adhesive joint:

Figure 13. Adhesive joint.

We suppose that there is a generalized plane strain e = 0 state. The linear force F value is 100MPa.mm. Taking the symmetries into account, the problem may be reduced to the lower half of the assembly (see Figure 13).

In order to perform the stress analysis in this assembly with the help of DEILAM, follow these stages:

1. Definition of the laminate (see Figure 14). In the main window of DEILAM, in the space of the total number of layers type 6. Then, in the table, enter the properties of the layers defined in Figure 13. Since these are isotropic material layers:
   • " The Young's moduli E_L , E_T and E_N value is E (the isotropic material Young's modulus)
   • " The Poisson's ratios n_LT and n_LN value are n (the isotropic material Poisson's ratio>>
   • " The shear moduli G_LT, G_LN and G_TN value is (the isotropic material shear modulus).

Figure 14. Defining the properties of the laminate.

2. Definition of the calculation properties. In the main window menu, click on <<Options>> of DEILAM. In the window that will appear, uncheck the 2 options of taking into account the thermal and hygroscopic strains (see Figure 15). Select the option all the generalized forces and displacements (see Figure 15).

Figure 15. Definition of the calculation properties.

3. Loads definition. In the main window of DEILAM, click on the button <<Loads>>. In the window that appears (called <<Configuration of the hygrothermal and mechanical loads>>), in the list of predetermined configurations select <<None>> (see Figure 16). Now, in this window, select in the configuration of the boundary conditions the options of the forces and moments on the left and right sides. Then, type 0 in the strain space e (see Figure 16). Then click on the button <<Values of the boundary conditions>>. A new window (called <<Values of the boundary conditions>>. In this new window, you define first the values at the left edge as follows (see Figure 17):

   In the type of data input, select the option input per layer and select layer 1. In the options of forces and displacements, select the <<Left>> edge and write zeros everywhere except for FY = 100MPa.mm. Then click on the button <<Save the values>> (you will see the input values in the table below the window). For layers 2 and 3 at the left edge, repeat the same operation without forgetting to specify the layer number and to write zeros for all of the forces and moments.

   For the right edge, the same instructions are followed except that the edge to be selected is the <<Right>> edge and you write zeros for all of the forces and moments in every layer except for the force FY = 100MPa.mm applied on layer 3.

   Finally, click on the button <<OK>> in order to validate the data and you will go to the window called <<Configuration of mechanical and hygrothermal load>>. You have to click on the button <<Read the configurations>>

Figure 16. First stage in the loads configuration.

Figure 17. Second stage in the loads configuration

4. Meshing definition. In the main window, click on the button <<Meshing of the width 0, b >>. In the window that appears, in the list of meshing types, select the option <<Automatic>>. In the space of the width value b, type 30.
5. Executing the calculations. Click on the button <<Analyze>> of the DEILAM main window.
6. Visualization of results. After the last stage, a new window appears. In this window, click on the button <<Fields to be displayed>>, and in the new window that appears, select in the list <<Fields to be displayed>> the option <<Interlaminar stresses>>. Check the box corresponding to the field (normal stresses) and put in the corresponding space 1-2 (in order to draw the stresses in interfaces 1 and 2: interfaces steel/adhesive and adhesive/steel, respectively). Then click on the button <<Draw>>. You will see a graphic similar to the one in Figure 18: it is the normal stresses curve at interfaces 1 and 2 vs. the position (y) on the interface. In order to distinguish the two curves, you only have to click on the button <<Chart properties>> and select the colors and types of lines you want for distinguishing the two curves.

Figure 18. First stage in the loads configuration.

7. Visualization of other fields. In order to visualize other fields such as membranar forces or rotations, you have to retake stage 6.

Determining the residual thermal stresses and displacements

Let us consider a symmetrical multi-layered material (q₁,...,q_n)_S made up of N = 2n layers of orthotropic materials. The multi-layered material is only subjected to a temperature variation DT. For this problem, there are no external forces applied to the laminate. So, the axial strain e should be such that the global force on the x direction:

is zero.

In this MAC LAM version, it is not possible to give the value of the force F. Despite this, it is possible to calculate the axial strain e that would give a null force F. In order to do this, you just have to follow these stages:

1. Calculation of the global stiffness R in the x axe (this stiffness takes into account the edge effect). This rigidity is calculated by performing first a calculation of the type <<free edge problem>> without considering the temperature and with a non-zero arbitrary axial strain e₁. Then you save the results of the in-plane forces (membranar forces) Nⁱ_xx for all of the layers and integrate the sum of all of the fields Nⁱ_xx along the width [0,b]; the result of the integration is a non-zero force F₁. Finally, is calculated.

R = F₁ / e₁

2. Intermediary calculation with a non-zero temperature variation DT and a zero axial strain e₂ = 0. Do not forget to enter the thermal expansion coefficients. After having finished the calculations, the results of the in-plane forces Nⁱ_xx are saved for all of the layers and the sum of all these fields Nⁱ_xx is integrated along the width [0,b]; the result of the integration is a force F₂.
3. Calculation of axial strain e which gives, for a temperature variation DT, a force F = 0. This strain is given by the formula:

e = -F₂ / R

After having determined the strain , the same stages are followed as in the case of the free edge problem, but this time with:

• An axial load e equal to the one calculated by means of the above-said formula and
• a temperature variation DT (do not forget to enter the thermal expansion coefficients).

The results obtained by means of this last calculation are the residual fields due to a single temperature variation.

Important remarks

References

[1] A. Diaz Diaz, J.F. Caron, R.P. Carreira, “Software application for evaluating interfacial stresses in inelastic symmetrical laminates with free edges”, Journal of Composite Structures, 58, 195-208, 2002

[2] CARREIRA RP, CARON JF, DIAZ DIAZ A. “Model of multi-layered materials for interface stresses estimation and validation by finite element calculations”, Mechanics of Materials, 34, 217-230, 2002.

[3] CARON JF, CARREIRA RP, DIAZ DIAZ A. “Critère d'initiation de délaminage dans les stratifiés”. Comptes Rendus de l'Académie des Sciences, 327, 1291-1296, 1999

[4] CARON JF, DIAZ DIAZ A, CARREIRA RP, CHABOT A, EHRLACHER A. “Multi-particle modelling for the prediction of delamination in multi-layered materials”. Artículo aceptado para publicación en la revista Composite Science and Technology, 2005.

[5] DIAZ DIAZ A, CARON JF. “Prediction of the onset of mode III delamination in carbon-epoxy laminates”. Artículo aceptado para publicación en la revista Composite Structures, 2005.