1. Numerical analysis of stochastic effective properties for polymer-based composites
M. Kamiński1,2, B. Lauke2
1 Department of Structural Mechanics, TU Łódź, Poland
2 Institute of Polymer Research, Dresden, Germany
The first author would like to acknowledge the financial support of the Research Grant from the Polish Ministry of the Research and Higher Education, NN 519  and the visiting professor fellowship from IPF Dresden, Germany.
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

2. Presentation outline Introduction and general assumptions
Available theories for effective shear modulus
Probabilistic and stochastic numerical techniques
Computational illustrations
Concluding remarks and further works
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

3. General assumptions
The composite consists of reinforcing particles (silica or carbon - filler) embedded into the elastomer (rubber + polymer matrix).
Generally filler has apparently larger elastic modulus than the elastomer.
All random events have predictable expectations but with some uncertainty level (not quite unpredictable).
Ageing according to the aggresive environment, extensive cyclic fatigue, moisture etc. and quite unpredictable events (like breakage) are omitted for a brevity.
Statistical moments analysis instead of the spectrum analysis or series of realizations.
Uncertainties as phase changes leading to the mathematical, mechanical or physical model changes are not implemented.
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

4. Available theories for effective shear modulus
Deformation and cluster size independent
Einstein-Smallwood
Guth-Gold
Pade approximation j
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

5. Available theories for effective elastic modulus
virgin elastomer S
Homogenization theory may depend on the specific constituents’ composition
carbon black
GH (Glassy Hard)
phase (2 nm)
SH (Sticky Hard)
phase (3-8 nm)
crosslinked rubber S
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

6. Available theories for effective shear modulus
Deformation independent but cluster size dependent
or generally
- primary particles and aggregates (below gel point)
- clusters (large concentration)
diameter d
backbone
inelastic end
cluster
aggregate
primary particles
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

7. Available theories for effective shear modulus
Deformation and cluster size dependent
exponential cluster breakdown
power-law cluster breakdown
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

8. Probabilistic and stochastic numerical techniques
b – input random variable (not necessarily Gaussian)
The central kth probabilistic moments for modulus G(eff)(b)
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

9. MCS & GSPT 50 x 104 x thomog 50 x 11 x thomog
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

10. The stochastic processes and ageing Biological populations
Technical systems
Gompertz-Makeham mortality law
Empirical power law
A – the age-independent term (Makeham parameter - extrinsic causes of death, such as accidents and acute infections),
B - the age-dependent term (the Gompertz function - due to deaths from age-related degenerative diseases like cancer and heart disease).
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

11. The stochastic ageing hypotheses Biological populations
Technical systems
Gompertz-Makeham mortality law
Empirical power law
for some physical constants c > 0, b > 0;
corrosion of reinforcement (linear: b = 1; Ellingwood & Mori (1993));
sulfate attack (parabolic: b = 2; Ellingwood & Mori (1993));
diffusion-controlled aging (square root: b = 0.5; Ellingwood & Mori (1993)),
creep (b = 1/8; Çinlar et al. (1977));
the expected scour-hole depth (b = 0.4; Hoffmans & Pilarczyk (1995) and van Noortwijk & Klatter (1999));
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

12. The stochastic ageing hypotheses
Consider the linear process with Gaussian coefficients
Stochastic ageing process is continuous with continuous time and non-stationary
The moments histories are computed instead of the trajectory verification
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

13. Computational illustrations Deformation and cluster size independent
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

14. Computational illustrations Deformation and cluster size independent
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

15. Computational illustrations
Deformation and cluster size dependent - exponential
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

16. Computational illustrations
Deformation and cluster size dependent - exponential
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

17. Computational illustrations
Deformation and cluster size dependent – power-law
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

18. Computational illustrations
Deformation and cluster size dependent – power-law
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

19. Computational illustrations
Deformation and cluster size dependent – stochastic case
Exponential model
Power-law model
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

20. Computational illustrations
Deformation and cluster size dependent – stochastic case
Exponential model
Power-law model
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

21. Computational illustrations
Deformation and cluster size dependent – stochastic case
Exponential model
Power-law model
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

22. Computational illustrations
Deformation and cluster size dependent – stochastic case
Exponential model
Power-law model
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

23. Concluding remarks
The coefficient f in most models is non-Gaussian, so that G(eff) also independently from G0.
Higher order perturbations in stochastic models result in no additional time consumption.
Successful validation of the perturbation method, which allows for up to 4th central probabilistic moments and coefficients.
Stochastic model leads to a decrease of the all characteristics for G(eff) in exponential and power-law models.
A complete set of stochastic data for the lifetime prediction and reliability analysis computed needs some further experiments (at least basic statistics for each design parameter).
Basic references
[1] M. Klϋppel, The role of disorder in filler reinforcement of elastomers on various length scales. Adv. Polym. Sci. (2003) 164: 1-86.
[2] Y. Fukahori, The mechanics and mechanism of the carbon black reinforcement of elastomers. Rubber Chem. Techn. (2004) 76:
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011

24. Further work - the stochastic FEM homogenization FEM discretization
homogenization function
homogenization tests
RVEs DOFs
18th International Conference on Composite Materials (ICCM18)
Jeju Island, South Korea, August 2011