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Journal Club Theme of October 2009: Peridynamics applied to the structure and evolution of discontinuities
Discontinuities have a fundamental role in the mechanics of solids. The most famous type of discontinuity is a crack, but others are important too, such as dislocations and phase boundaries. Many types of deformation that appear to be continuous at the macroscale, such as plastic flow in metals, really involve the evolution of discontinuities at some smaller scale.
How should we model discontinuities within a continuum framework? This is a long-standing question. Let’s first focus on cracks. The PDEs of the standard theory of solid mechanics cannot be applied directly at the points in a continuous body along which a crack grows, because the necessary spatial derivatives of the deformation do not exist there. (Using the weak form of the PDEs does not resolve this issue.) Therefore, the standard approach to modeling cracks is to introduce extra equations into the standard theory. A separate kinetic relation, extraneous to the basic field equations, is supplied to determine how a crack grows. This approach brings with it the following challenges and questions, among others:
- What determines whether a crack should nucleate, i.e., what mathematical conditions lead to the spontaneous appearance of a discontinuity within a previously continuous body?
- How can we determine, in sufficient generality, the correct kinetic relation that governs crack motion? How fast does a crack grow? In what direction should it grow? Should it branch? Should it change mode? Should it oscillate? Should it arrest? What should all these things depend on? How do we account for interfaces and defects close enough to the crack tip to affect the asymptotic fields? The science of fracture mechanics is largely concerned with these questions, but the answers it provides are sometimes limited to ideal conditions.
- In a computational method designed to solve the standard PDEs in a continuous body, how do we keep track of all the new surfaces created by crack growth? How do we ensure that the discretization does not artificially introduce preferred directions for crack growth? How do we know that the method converges to some exact solution?
The peridynamic formulation of solid mechanics attempts to address these issues by replacing the standard PDEs with new field equations that, we hope, are better suited to the study of discontinuities. These field equations, which are integro-differential equations, can be applied directly on discontinuities. Cracks nucleate, initiate, and grow spontaneously according to the equation of motion and constitutive model, which do not involve the spatial derivatives of the deformation. Cracking is treated as just another form of deformation rather than as a separate phenomenon that requires separate equations. Cracks “do whatever they want.”
I would like to refer to four papers that demonstrate the potential of the theory and some of its current limitations. The first of these papers demonstrates unguided crack growth in a peridynamic solid. In the problems studied in this paper, the driving force for crack growth comes from a temperature gradient.
- B. Kilic and E. Madenci, Prediction of crack paths in a quenched glass plate by using peridynamic theory, International Journal of Fracture 156 (2009)165–177 (http://dx.doi.org/10.1007/s10704-009-9355-2)
This next paper applies the peridynamic method to composite laminates. It demonstrates the influence of fiber directions on the direction of crack growth in the presence of strongly anisotropic properties:
- B. Kilic, A. Agwai, and E. Madenci, Peridynamic theory for progressive damage prediction in center-cracked composite laminates, Composite Structures 90 (2009) 141-151 (http://dx.doi.org/10.1016/j.compstruct.2009.02.015)
The next paper considers the effect of long-range forces on the failure of a complex heterogeneous body at the nanoscale.
- F. Bobaru, Influence of van der Waals forces on increasing the strength and toughness in dynamic fracture of nanofibre networks: a peridynamic approach, Modelling and Simulation in Materials Science and Engineering 15 (2007) 397–417 (http://dx.doi.org/10.1088/0965-0393/15/5/002).
The following paper applies the method not to cracks, but to martensitic phase boundaries. It demonstrates that a kinetic relation for the motion of a phase boundary is implied by the peridynamic field equations. The method also predicts certain structural features within a phase boundary, as well as complexity in the evolution of phase boundaries in multiple dimensions.
- K. Dayal and K. Bhattacharya, Kinetics of phase transformations in the peridynamic formulation of continuum mechanics, Journal of the Mechanics and Physics of Solids 54 (2006) 1811–1842 (http://dx.doi.org/10.1016/j.jmps.2006.04.001).
The capabilities of the peridynamic method, particularly the generality of material response that can be modeled within it, are evolving rapidly. Although the peridynamic model has pros and cons that can be debated, I hope that the community will consider the basic question that it tries to address to be worthy of discussion: Is the standard PDE-based continuum theory the best possible tool for modern solid mechanics?