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Length Scales in Fracture
Some material scientists and experimentalists are generally sceptical of simulations and reproach the theoreticians with lacking knowledge of real materials. Sometimes they may just be ignorant of the mathematics behind the models but sometimes they appear to be right. An example: Introducing “damage” into a constitutive equation simply as an internal variable which obeys some evolution law, not having the foggiest notion about the specific nature of damage - whether brittle, ductile, creep, fatigue - and its micromechanical mechanisms, promotes scepticism about the benefits of modelling in general, and deservedly so.
This is not my particular point today, however, but it is related. My problem today is the handling of length scales in
Ductile tearing is governed by the initiation, growth and coalescence of voids in an elasto-plastic or viscoplastic material. Koplik and Needleman (1981) have been the first to perform unit-cell calculations of void growth to analyse this mechanism, and numerous studies by other authors followed varying the void shape, accounting for inclusions etc.. They helped improving and generalising constitutive equations of ductile damage like the Gurson model of porous metal plasticity. Unit-cell calculations have also been used to derive traction-separation laws for cohesive zones. The physical processes take place at length scales of micrometers to millimetres, accordant to the dimensions of the microstructure. Continuum models still apply at this length scale.
Atomistic simulations and molecular dynamics are based on models that relate binding energies or forces to spatial configurations in order to calculate accelerations of particles via Newton’s law. They describe interactions at a length scale of nanometers, which is at least three orders of magnitude below the relevant length scale of ductile tearing. What occurs at this length scale has absolutely nothing in common with plastic deformations due to dislocation motion and ductile tearing of metals, and hence the authors’ conclusions are apocryphal and unsubstantiated:
> “The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale.”
What do the authors actually mean by vague formulations like “in the form of void nucleation ...” and “similar to ductile fracture”? Voids nucleate at particles, for instance. Which particles are of atomic or sub-atomic size? What is their criterion for a process being “similar to” another?
> “Understanding the failure process based on atomistic simulation can provide detailed information for the cohesive law.”
This is correct, of course, provided the correct failure process is modelled.
> “The relationship between atomic traction and atomic separation including elastic deformations confirms the exponential function form suggested by Needleman.”
This is neither a miracle nor striking news, as the cohesive law proposed by Needleman in Int. J. Fracture 42 (1990) is based on the universal atomistic binding energy function proposed by Rose et al. in Phys. Rev. Letters 47 (1981), and he actually analysed “tensile decohesion along an interface” in an elastic medium.
> “The computations show that void nucleation and growth are controlled by tensile stress and hydrostatic stress. The Mises stress is not involved in material failure.”
This is the final death sentence to any simulation: that it yields results which are contrary to everything that is known about the real process. Apart from this, the hydrostatic stress and the maximum tensile stress depend on each other under fully plastic conditions of plane strain. But plasticity is out of the scope of the MD simulations, anyway.
After all, the simulations could indeed provide useful information on a cohesive law for a process which is not void growth and coalescence in ductile metals, however. It is the authors’ business to present an actual decohesion mechanism following their model.