8 Damage

The first and most common mode of damage is matrix cracking. Laminate matrix cracking can be induced by transverse strain, shear strain, or both; as well as by impact. CADEC can predict this type of damage for symmetric laminates subjected to any combination of membrane strains. A generalization for unsymmetric laminates and bending deformations is being worked out.

Damage analysis is accessed from Chapters on the menu tree, which displays a page like this:
\includegraphics[]{./Images/Damage.png}

Only symmetric laminates will be shown in the Symmetric Laminate selection box, so make sure you use the symmetric checkbox when you define the laminates in My Documents.

Then you choose the environment. The predictive model, called DDM, accounts for temperature but not for moisture yet.

$\Delta T$ is the temperature deviation from the temperature at which the material properties $G_{IC},G_{IIC}$ were measured. These properties (critical energy release rates) are entered in My Documents for each lamina in the laminate. If experimenal values are not available, they can be estimated as explained in Intrinsic lamina properties.

Next you provide the load in the form of a set of applied strains. A default of 20,000 micro-strains (0.02 strain = 2% strain) is placed on the page for your convenience, but you can use any combination of strains you like. Internally, the DDM algorithm [5] divides your input into 60 steps and calculates the crack density in each lamina for each increment of strain. The result are provided in and Excel file like Table 8.1.

$\epsilon _ x$

$\epsilon _ y$

$\epsilon _{xy}$

$\lambda _1$

$\lambda _2$

D22 (1)

D22 (2)

D66 (1)

D66 (2)

0

0

0

0.02

0.02

0.144

0.144

0.1203

0.1236

0.0003

0

0

0.02

0.02

0.144

0.144

0.1237

0.1237

0.0007

0

0

0.02

0.02

0.144

0.144

0.1237

0.1237

0.001

0

0

0.02

0.02

0.144

0.144

0.1237

0.1237

0.0014

0

0

0.02

0.02

0.144

0.144

0.1237

0.1237

0.0017

0

0

0.02

0.02

0.144

0.144

0.1237

0.1237

0.002

0

0

0.02

0.0447

0.144

0.3077

0.1283

0.2557

0.0024

0

0

0.02

0.0654

0.144

0.4318

0.1327

0.3522

0.0027

0

0

0.02

0.0815

0.144

0.517

0.1364

0.4196

0.0031

0

0

0.02

0.096

0.144

0.5832

0.14

0.4746

0.0034

0

0

0.02

0.1089

0.144

0.6345

0.1435

0.5197

...

0.02

0

0

0.02

0.4549

0.144

0.9633

0.2652

0.9294

Table 8.1: DDM output for Example 7.5. Click the Back button on your browser to return to your previous page.

Here, $\epsilon _ x,\epsilon _ y,\epsilon _{xy}$ are the incrementally applied strains, which by step 60 coincide with the user input values of strain.

Then, $\lambda _1,\lambda _2$ are the crack densities, in cracks/mm, in laminas 1 and 2, respecively. You can see that at the first step, with zero applied strains, the crack density is already 0.02 for all laminas. This is because DDM needs an initial damage to be seeded in the model of the actual laminate. This seeding represents the initial defects that always exists on the material. Studies have shown that the results are insensitive to the seed used as long it is small. A seed crack density of 0.02 represents one crack every 50 mm. We could reduce it to 0.01 or even less but the only difference would be slightly more computer time to get the results. For an online application such as CADEC, fast calculation time is important, because the user has to wait for the page to update, so we picked 0.02 as a good compromise between accuracy and speed.

Notice that for this example, lamina 1 is a 0-deg lamina, loaded longitudinally with $\epsilon _ x$, so most of the load in lamina 1 is carried by the fibers, and thus it does not crack. Its crack density remains at 0.02. On the other hand, lamina 2 is a 90-deg lamina, loaded transversely. Crack initiation happens at strain 0.002, jumping from crack density 0.02 to 0.0447 cracks/mm. Crack evolution (growth) takes place from then on, reaching 0.4549 cracks/mm when the applied strain reaches 0.02. The crack density evolution (history) is shown below:

\includegraphics[width=\textwidth ]{./Images/FigureDamageExample75.png}
Figure 8.1: DDM output plotted in Excel for Example 7.5. Click the Back button on your browser to return to your previous page.

Notice how cracks start at strain=0.0020, while the last point with crack density equal to the seed is 0.0017. Crack density is predicted to grow with applied strain as shown in Figure 8.1.

The damage parameters $D_{22}, D_{66}$ represent the loss of stiffness. DDM predicts the loss of stiffness for the laminate and for each lamina as a function of predicted crack density in the laminas. Then, the damage parameters are calculated as follows [2, Eq. (8.39)]:

  $\displaystyle  D_{ij}^ k=1-{Q_{ii}^ k}/{\widetilde Q_{ii}^ k}\notag  $    

where k is the lamina number, $\widetilde Q^ k, Q^ k$ are the undamaged and damaged (degraded) stiffness of lamina k, respectively, and i=2,6 point to the transverse strain and in-plane shear strain, respectively. A value D=1 means that the stiffness of that lamina has been lost completely.

The analysis conducted by CADEC is for a laminate subjected to the strains given on the page, applied uniformly to the entire laminate. To analyze damage of a complex part, such as plate with a hole [9], you need DDM running inside a finite element package. Such tool is available for ANSYS$^{\rm R}$[4] and Abaqus$^{\rm TM}$[3]. An example output obtained with Abaqus$^{\rm TM}$ is shown below:

\includegraphics[width=\textwidth ]{./Images/FigureMoure13.png}
Figure 8.2: Evolution of crack density with increasing load on a plate with a hole [9]. Click the Back button on your browser to return to your previous page.