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# JOURNAL CLUB DEC 1 08: CYBER-ENABLED PREDICTIVE SCIENCE-BASED CONTINUUM MECHANICS FOR MULTISCALE FRACTURE PROCESS DISCOVERY

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The key to developing materials against failure under extreme environments is the understanding of the interplay between the various physical scales present, from the atomic level interactions, to the microstructural composition and the macroscale behavior. The ability to do this requires new methods that can be used to predict macroscale properties accurately based on microstructures and nanostructures without resorting to empiricism.

The relationship between macroscale properties and the underlying microstructure is often the most difficult to define, as microstructure is often in an evolving state during the lifetime of a component. Imaging techniques such as X-ray Computed Tomography, light microscopy, scanning electron microscopy, transmission electron microscopy and local electrode atom probe can be used to accurately characterize the microstructure at a snapshot in time. The material properties, which are determined from physical experiments, can then be related to the observed microstructure.

Predicting microstructure-property relationships in engineering materials via direct simulation of the underlying micromechanics remains an elusive goal due to the massive disparities in length and time scales; huge structures such as bridges may fail after decades of operation due to microscale fracture events which occur at a micro/nano second time scale. Even as we move into the era of petascale computing, affordability remains a key factor when developing a computational method.

In an effort to bypass the limitations of computational speed and storage capacity, scientists and engineers in many fields have focused on meshfree and particle methods [1-4] for large deformation analysis, multiscale analysis theories for coupling atomistic and continuum [5,6], and hierarchical and concurrently multiscale methods [7-9].

The proposed research entitled “*Cyber-Enabled Predictive Science-Based Continuum mechanics (PSBCM) for Multiscale Fracture Process Discovery” *will overcome the limitations of conventional higher order/gradient continuum models and direct coupling schemes. The key features of the proposed framework are [10-11]:

- Allows macroscale properties and performance to be
**predicted**directly in terms of the key microstructure parameters including length scales of inhomogeneous deformation. - Provides a foundation for microstructure-level computational materials and design of structural components for multiscale shear fracture.

In particular the proposed PSBCM will work as follows:

- Predicts the evolving scale and magnitude of inhomogeneous deformation directly in terms of the evolving microstructural parameters and length scales.
- Employs a set of continuum microstresses which describe the resistance to inhomogeneous deformation at each characteristic length scale in the evolving microstructure. These microstresses are directly coupled to the conventional macroscale stress through a set of multiresolution continuum governing equations.
- The time dependent microstructural continuum governing equations can be discretized and
*solved using a conventional finite element analysis approach with a single mesh*. - Does not require complicated inter-scale boundary conditions.

The proposed theory is based on a *multiresolution (multiscale) homogenization* approach. The continuum governing equations [7-13] are developed by averaging over a set of *nested RVE* (representative volume element) *at several scales within the microstructured RVEs associated with its inhomogeneous deformations*. The result is a set of multiresolution continuum partial differential governing equations which are a continuation of the continuum governing equations involving extra *inhomogeneous microstresses within a material continuum point*. These microstresses arise naturally from the extra small scale averaging operations within the RVE. Conveniently, a traditional finite element solution procedure may be used to discretize and solve the weak form of the multiresolution governing equations.

In terms of constitutive model development, average constitutive relationships are now required at each scale of interest within the RVE to describe the extra *inhomogeneous* microstresses in the extended multiresolution governing equations. The constitutive relationships are *derived* by averaging the stress and strain at each scale within *computational nested RVE models*, resulting in a set of *concurrently coupled generalized statistically described stress-strain relation*. Hence, we are able to describe the macroscopic response in terms of the underlying microstructure physics via the multiresolution governing continuum equations and the generalized stress-strain law.

We are particularly interested in predicting the macroscale constitutive characterization for the dynamic multiscale fracture processes and its propagation in which the rate and strain hardening stabilizing effects are overcome by a combination of microvoid nucleation at several scales and hence, thermal softening. We are also interested in how microstructures evolve spatially and temporally, and relating an evolving microstructure to macroscale properties and performance.

**Selected discussion topics:** maturity of multiscale methods; role of petascale computing multiscale materials design; what is the role of predictive science-based engineering, i.e., science-based continuum mechanics; multiscale experiments for validations; incorporation of uncertainties and defects; verification, validation and uncertainty quatification (V&V and UQ); training of the next generation of students.

**References**

1. W. K. Liu et al., *Reproducing Kernel Particle Methods*, International Journal for Numerical Methods in Fluids, 20, 1081-1106, 1995.

2. W. K. Liu et al., *Reproducing Kernel Particle Methods for Structural Dynamics*, International Journal for Numerical Methods in Engineering, 38, 1655-1679, 1995.

3. Li, S., and Liu, W. K., *Meshfree and Particle Methods and Their Applications*, Applied Mechanics Review, 55, 1-34, 2002.

4. Shaofan Li and Wing Kam Liu, *Meshfree Particle Methods*, Springer, 502pp. 2004.

5. Gregory J. Wagner and Wing Kam Liu, *Coupling of Atomic and Continuum Simulations Using a Bridging Scale Decomposition*, Journal of Computational Physics, vol. 190, pp. 249-274, 2003.

6. Qian, D. et al., *A Multiscale Projection Method for the Analysis of Carbon Nanotubes*, Computer Methods in Applied Mechanics and Engineering, 193(17-20), 1603-1632, 2004.

7. W.K. Liu et al., *An Introduction to Computational Nanomechanics and Materials*, Computer Methods in Applied Mechanics and Engineering, 193, 1529–1578, 2004.

8. W.K. Liu, E. Karpov, H. Park, *Nano Mechanics and Materials: Theory, Multiscale Methods and Applications*, John Wiley and Sons, 2005.

9. Qian, D. et al,, *Concurrent quantum/continuum coupling analysis of nanostructures*, Computer Methods in Applied Mechanics and Engineering. 197(41-42), 3291-3323, 2008.

10. F. J. Vernerey et al., *Multiscale Micromorphic Theory for Hierarchical Materials,* Journal of the Mechanics and Physics of Solids, 55(12), 2603-2651, 2007.

11. F. J. Vernerey et al., *A Micromorphic Model for the Multiple Scale Failure of Heterogeneous Materials,* Journal of the Mechanics and Physics of solids, 56(4), 1320-1347, 2008.

12. Wing Kam Liu and Cahal McVeigh. *Predictive Multiscale Theory for Design of Heterogeneous Materials,* Computational Mechanics, 42(2), 147-170, 2008.

13. Cahal McVeigh and Wing Kam Liu, *Linking microstructure and properties through a predictive multiresolution continuum*, Computer Methods in Applied Mechanics and Engineering, 197, 3268–3290, 2008.

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