A Stochastic Framework for Atomistic Fracture
Pages 526 - 548
Abstract
We present a stochastic modeling framework for atomistic propagation of a Mode I surface crack, with atoms interacting according to the Lennard--Jones interatomic potential at zero temperature. Specifically, we invoke the Cauchy--Born rule and the maximum entropy principle to infer probability distributions for the parameters of the interatomic potential. We then study how uncertainties in the parameters propagate to the quantities of interest relevant to crack propagation, namely, the critical stress intensity factor and the lattice trapping range. For our numerical investigation, we rely on an automated version of the so-called numerical-continuation enhanced flexible boundary NCFlex algorithm.
References
[1]
E. Bitzek, J. R. Kermode, and P. Gumbsch, Atomistic aspects of fracture, Int. J. Fracture, 191 (2015), pp. 13--30.
[2]
J. Braun, M. Buze, and C. Ortner, The effect of crystal symmetries on the locality of screw dislocation cores, SIAM J. Math. Anal., 51 (2019), pp. 1108--1136.
[3]
M. Buze, T. Hudson, and C. Ortner, Analysis of an atomistic model for anti-plane fracture, Math. Models Methods Appl. Sci., 29 (2019), pp. 2469--2521.
[4]
M. Buze, T. Hudson, and C. Ortner, Analysis of cell size effects in atomistic crack propagation, ESAIM Math. Model. Numer. Anal., 54 (2020), pp. 1821--1847.
[5]
M. Buze and J. R. Kermode, Numerical-continuation-enhanced flexible boundary condition scheme applied to mode-i and mode-iii fracture, Phys. Rev. E, 103 (2021), 033002.
[6]
W. A. Curtin, On lattice trapping of cracks, J. Mater. Res., 5 (1990), pp. 1549--1560.
[7]
W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), pp. 241--297.
[8]
J. L. Ericksen, On the Cauchy-Born rule, Math. Mech. Solids, 13 (2008), pp. 199--220.
[9]
J. L. Ericksen, The Cauchy and Born hypotheses for crystals, in Phase Transformations and Material Instabilities in Solids, M. E. Gurtin, ed., 1984, pp. 61--77.
[10]
S. L. Frederiksen, K. W. Jacobsen, K. S. Brown, and J. P. Sethna, Bayesian ensemble approach to error estimation of interatomic potentials, Phys. Rev. Lett., 93 (2004), 165501.
[11]
G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice, J. Nonlinear Sci., 12 (2002), pp. 445--478.
[12]
J. Guilleminot, Modeling non-Gaussian random fields of material properties in multiscale mechanics of materials, in Uncertainty Quantification in Multiscale Materials Modeling, Y. Wang and D. L. McDowell, eds., Elsevier Ser. Mech. Adv. Mat., Woodhead Publishing, Sawston, UK, 2020, pp. 385--420.
[13]
J. Guilleminot and C. Soize, On the statistical dependence for the components of random elasticity tensors exhibiting material symmetry properties, J. Elasticity, 111 (2013), pp. 109--130.
[14]
J. Guilleminot and C. Soize, Non-Gaussian random fields in multiscale mechanics of heterogeneous materials, Encyclopedia Contin. Mech., H. Altenbach and A. Öchsner, eds., Springer, Berlin, 2020, pp. 1826--1834.
[15]
P. Hänggi, P. Talkner, and M. Borkovec, Reaction-rate theory: Fifty years after Kramers, Rev. Modern. Phys., 62 (1990), pp. 251--341.
[16]
E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev., 106 (1957), pp. 620--630.
[17]
J. R. Kermode, T. Albaret, D. Sherman, N. Bernstein, P. Gumbsch, M. C. Payne, G. Csányi, and A. De Vita, Low-speed fracture instabilities in a brittle crystal, Nature, 455 (2008), pp. 1224--1227.
[18]
C. Kittel, P. McEuen, and P. McEuen, Introduction to Solid State Physics, 8th ed., Wiley, New York, 1996.
[19]
L. D. Landau, E. M. Lifshitz, and J. B. Sykes, Theory of Elasticity, Course Theoret. Phys., Pergamon Press, Oxford, 1989.
[20]
J. E. Lennard-Jones, On the determination of molecular fields. III. From crystal measurements and kinetic theory data, Proc. A, 106 (1924), pp. 709--718.
[21]
S. Longbottom and P. Brommer, Uncertainty quantification for classical effective potentials: An extension to potfit, Model. Simul. Mater. Sci. Eng., 27 (2019), 044001.
[22]
M. M. Mehrabadi and S. C. Cowin, Eigentensors of linear anisotropic elastic materials,
Quart. J. Mech. Appl. Math., 43 (1990), pp. 15--41.
[23]
M. L. Mehta, Random Matrices, Elsevier, New York, 2004.
[24]
L. A. Mihai, T. E. Woolley, and A. Goriely, Stochastic isotropic hyperelastic materials: Constitutive calibration and model selection,
Proc. A, 474 (2018), 20170858.
[25]
C. Ortner and F. Theil, Justification of the Cauchy-Born approximation of elastodynamics, Arch. Ration. Mech. Anal., 207 (2013), pp. 1025--1073.
[26]
M. Ostoja-Starzewski, Lattice models in micromechanics, Appl. Mech. Rev., 55 (2002), pp. 35--60.
[27]
J. E. Sinclair, The influence of the interatomic force law and of kinks on the propagation of brittle cracks, Philos. Mag., 31 (1975), pp. 647--671.
[28]
C. Soize, Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 26--64.
[29]
C. Soize, Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering, Inter. Disp. Appl. Springer, Berlin, 2017.
[30]
C. T. Sun and Z.-H. Jin, Fracture Mechanics, Academic Press, New York, 2012.
[31]
R. Thomson, C. Hsieh, and V. Rana, Lattice trapping of fracture cracks, J. Appl. Phys., 42 (1971), pp. 3154--3160.
[32]
M. F. Thorpe and I. Jasiuk, New results in the theory of elasticity for two-dimensional composites, Proc. Math. Phys. Sci., 438 (1992), pp. 531--544.
[33]
R. Veltz, BifurcationKit.jl, https://hal.archives-ouvertes.fr/hal-02902346, 2020.
[34]
M. Wen and E. B. Tadmor, Uncertainty quantification in molecular simulations with dropout neural network potentials, npj Comput. Mater., 6 (2020), pp. 1--10.
[35]
M. Wen, S. M. Whalen, R. S. Elliott, and E. B. Tadmor, Interpolation effects in tabulated interatomic potentials, Model. Simul. Mater. Sci. Eng., 23 (2015), 074008.
[36]
A. T. Zehnder, Fracture Mechanics, Springer, Berlin, 2012.
Recommendations
Phase-field modeling of ductile fracture
Phase-field modeling of brittle fracture in elastic solids is a well-established framework that overcomes the limitations of the classical Griffith theory in the prediction of crack nucleation and in the identification of complicated crack paths ...
Phase-field modelling and analysis of rate-dependent fracture phenomena at finite deformation
AbstractFracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack ...
Modeling orthotropic elasticity, localized plasticity and fracture in trabecular bone
This work develops a model for the mechanical response of trabecular bone including plasticity, damage and fracture. It features a resultant lamellar orientation that captures trabecular strut anisotropic elasticity, and introduces asymmetric J2 ...
Comments
Information & Contributors
Information
Published In
© 2022, Society for Industrial and Applied Mathematics.
Publisher
Society for Industrial and Applied Mathematics
United States
Publication History
Published: 01 January 2022
Author Tags
Author Tags
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 09 Jan 2025