2. 1 Correlation in elastic properties of composite laminates due to variability in fiber volume fraction Based on PhD defense slides of Benjamin P. Smarslok Correlation model for composite elastic properties using only measurements from a single specimen (in Press, Probabilistic Engineering Mechanics).

3. Composite Laminate Elastic properties Composite lamina properties: Properties may have substantial correlation, but correlation matrix may not be available Correlation may be able to be deduced from dependence on fiber and matrix properties Here variability in fiber volume fraction is propagated to correlation between elastic constants 1 2 E 1, E 2, G 12, 12,

4. 3 Material Variability and Measurement Error in Composite Properties Mechanisms for composite property variability: –Fiber misalignment –Fiber packing –Fiber volume fraction Develop a correlation model for composite material variability based on fiber volume fraction, V f Consider S-glass/epoxy and carbon fiber/epoxy (IM7/977-2) laminates –Combine with available variance-covariance measurement error data How do correlated uncertainties in composite properties propagate to strain or probability of failure?

5. 4 Composite Properties vs. V f Trends for a typical graphite/epoxy (applied to IM7/977-2) Fiber volume fraction V f (Figures from Caruso and Chamis 1986 and Rosen and Dow 1987)

6. Uncertainty model – Measurement error – Material variability –Composite properties: –Ex: Uncertainty model for E 2 : –Total variance - covariance: Measurement error is usually quantified or estimated, however variability data is often unavailable Use fiber volume fraction to approximate variability 5 Correlated Data for Material Properties

7. 6 Combining Uncertainty Combing material variability and measurement error: (Correlated)(Independent) Is it right to add covariance matrices?

8. 7 Vibration test identification results Use experimental data and combine with correlated material variability Glass/Epoxy: Measurement Uncertainty with Correlated Data (Gogu et al. - SDM2009) E1E1 E2E2 12 G 12 E1E1 3.5e18-0.14-0.38-0.63 E2E2 -3.1e171.4e18-0.59-0.36 12 -2.3e7 1.1e-30.77 G 12 -6.9e17-2.5e171.5e73.5e17 Material Property MeanStandard Deviation ( stdev ) Coefficient of Variation (CV) E 1 (GPa)611.93.1% E 2 (GPa)211.25.5% 12 0.270.0312.2% G 12 (GPa)100.596.0% covariance - correlation

9. 8 Glass/Epoxy Graphite/Epoxy Material Property MeanCoefficient of Variation (CV) E 1 (GPa)614.5% E 2 (GPa)218.9% 12 0.2712.3% G 12 (GPa)9.99.2% Material Property MeanCoefficient of Variation (CV) E 1 (GPa)1504.4% E 2 (GPa)94.1% 12 0.343.4% G 12 (GPa)4.66.0% Summary - Comparing Total Uncertainties E1E1 E2E2 12 G 12 E1 E1 7.5e180.52-0.350.29 E2 E2 2.7e183.5e18-0.460.46 12 -3.2e7-2.8e71.1e-30.39 G 12 7.2e177.8e171.2e68.3e17 E1E1 E2E2 12 G 12 E1 E1 4.3e190.66-0.440.84 E2 E2 1.6e181.3e17-0.300.59 12 -3.3e7-1.3e61.3e-4-0.39 G 12 1.5e186.0e16-1.2e67.7e16

10. 9 Independent vs. Correlated Material Properties Illustrative example: Propagate uncertainty to strain Ex: Cylindrical pressure vessel – NASAs X-33 RLV –4 layer (±25°) s P = 100kPa ( 50kPa) d = 1m t = 125 m Compare glass/epoxy and graphite/epoxy Max Strain Failure Criterion (deterministic)

11. 10 Monte Carlo simulations (10 5 ) with correlated and independent properties Ignoring correlations can cause unsafe or inefficient designs! Is the difference in means significant? Comparison of Failure Probability R.V. type mean( 2 ) CV( 2 ) pfpf mean( 2 )CV( 2 ) pfpf independent 1399 8.5%0.010 1245 3.2%0.007 correlated 1402 7.0%0.005 1246 4.1%0.026 Independent vs. Correlated SYM Glass/EpoxyGraphite/Epoxy

12. 11 Correlation Model Summary Uncertainty and correlation in composite properties: – Material variability – not always available Used information from fiber volume fraction – Measurement error – usually quantified or estimated 1.Correlated data 2.Experimental estimates Combined uncertainties in a general covariance model –Correlations dont need to be avoided! Neglecting correlations by using independent RVs can result in an inefficient or unsafe design!