Abstract
Using coarse graining, the upscaled mechanical properties of a solid with small scale heterogeneities are derived. The method maps internal forces at the small scale onto peridynamic bond forces in the coarse grained mesh. These upscaled bond forces are used to calibrate a peridynamic material model with position-dependent parameters. These parameters incorporate mesoscale variations in the statistics of the small scale system. The upscaled peridynamic model can have a much coarser discretization than the original small scale model, allowing larger scale simulations to be performed efficiently. The convergence properties of the method are investigated for representative random microstructures. A bond breakage criterion for the upscaled peridynamic material model is also demonstrated.
Similar content being viewed by others
Availability of Data and Material
Not applicable.
References
Peerlings R, Fleck N (2004) Computational evaluation of strain gradient elasticity constants. Int J Multiscale Comput Eng 2(4):599–619
Willis JR (1983) The overall elastic response of composite materials. J Appl Mech 50(4b):1202–1209
Torquato S (2005) Random heterogeneous materials: microstructure and macroscopic properties. Springer Science & Business Media
Buryachenko V (2007) Micromechanics of heterogeneous materials. Springer Science & Business Media
Milton GW (2022) The theory of composites. SIAM
Hashin Z, Shtrikman S (1962) On some variational principles in anisotropic and nonhomogeneous elasticity. J Mech Phys Solids 10(4):335–342
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140
Drugan WJ, Willis JR (1996) A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J Mech Phys Solids 44(4):497–524
Silling SA (2014) Origin and effect of nonlocality in a composite. J Mech Mater Struct 9(2):245–258
Alali B, Lipton R (2012) Multiscale dynamics of heterogeneous media in the peridynamic formulation. J Elast 106:71–103
Mehrmashhadi J, Chen Z, Zhao J, Bobaru F (2019) A stochastically homogenized peridynamic model for intraply fracture in fiber-reinforced composites. Compos Sci Technol 182:107770
Chen Z, Niazi S, Bobaru F (2019) A peridynamic model for brittle damage and fracture in porous materials. Int J Rock Mech Min Sci 122:104059
Wu P, Zhao J, Chen Z, Bobaru F (2020) Validation of a stochastically homogenized peridynamic model for quasi-static fracture in concrete. Eng Fract Mech 237:107293
Wu P, Yang F, Chen Z, Bobaru F (2021) Stochastically homogenized peridynamic model for dynamic fracture analysis of concrete. Eng Fract Mech 253:107863
Madenci E, Barut A, Phan ND (2017) Peridynamic unit cell homogenization. In: 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, number AIAA 2017-1138
Madenci E, Barut A, Phan N (2018) Peridynamic unit cell homogenization for thermoelastic properties of heterogenous microstructures with defects. Compos Struct 188:104–115
Hu Y, Wang J, Madenci E, Mu Z, Yu Y (2022) Peridynamic micromechanical model for damage mechanisms in composites. Compos Struct 301:116182
Xia W, Oterkus E, Oterkus S (2021) Ordinary state-based peridynamic homogenization of periodic micro-structured materials. Theoret Appl Fract Mech 113:102960
Buryachenko VA (2014) Some general representations in thermoperistatics of random structure composites. Int J Multiscale Comput Eng 12(4):331–350
Buryachenko VA (2017) Effective properties of thermoperistatic random structure composites: some background principles. Math Mech Solids 22(6):1366–1386
Buryachenko VA (2020) Variational principles and generalized Hill’s bounds in micromechanics of linear peridynamic random structure composites. Math Mech Solids 25(3):682–704
Buryachenko VA (2020) Effective deformation of peridynamic random structure bar subjected to inhomogeneous body-force. Int J Multiscale Comput Eng 18(5):569–585
Buryachenko VA (2020) Generalized effective fields method in peridynamic micromechanics of random structure composites. Int J Solids Struct 202:765–786
Buryachenko VA (2023) Effective displacments of peridynamic heterogeneous bar loaded by body force with compact support. Int J Multiscale Comput Eng 21(1):27–42
Buryachenko VA (2023) Estimations of effective energy-based criteria in nonlinear phenomena in peridynamic micromechanics of random structure composites. J Peridynamics Nonlocal Model. https://doi.org/10.1007/s42102-023-00096-7
Buryachenko VA (2023) Linearized ordinary state-based peridynamic micromechanics of composites. J Mech Mater Struct 18(4):445–477
Buryachenko VA (2023) Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems. Math Mech Solids 28(6):1401–1436
Silling SA (2011) A coarsening method for linear peridynamics. Int J Multiscale Comput Eng 9(6):609–622
You H, Yu Y, Silling S, D’Elia M (2022) A data-driven peridynamic continuum model for upscaling molecular dynamics. Comput Methods Appl Mech Eng 389:114400
You H, Xu X, Yu Y, Silling S, D’Elia M, Foster J (2023) Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures. Preprint at http://arxiv.org/abs/2301.04540
Xu X, D’Elia M, Foster JT (2021) A machine-learning framework for peridynamic material models with physical constraints. Comput Methods Appl Mech Eng 386:114062
You H, Yu Y, Silling S, D’Elia M (2020) Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws. Preprint at http://arxiv.org/abs/2012.04157
Fan Y, D’Elia M, Yu Y, Najm HN, Silling S (2023) Bayesian nonlocal operator regression: a data-driven learning framework of nonlocal models with uncertainty quantification. J Eng Mech 149(8):04023049
Fan Y, Tian X, Yang X, Li X, Webster C, Yu Y (2022) An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems. J Comput Phys 465:111376
Fan Y, You H, Tian X, Yang X, Li X, Prakash N, Yu Y (2022) A meshfree peridynamic model for brittle fracture in randomly heterogeneous materials. Comput Methods Appl Mech Eng 399:115340
Madenci E, Oterkus E (2013) Peridynamic Theory and Its Applications. Springer, New York
Silling SA, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184
Silling S, Littlewood D, Seleson P (2015) Variable horizon in a peridynamic medium. J Mech Mater Struct 10(5):591–612
Silling SA, Adams DP, Branch BA (2023) Mesoscale model for spall in additively manufactured 304L stainless steel. Int J Multiscale Comput Eng 21(3):49–67
Acknowledgements
This article has been authored by an employee of National Technology and Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan.
Funding
Funding was provided by the U.S. Department of Energy through the Advanced Certification and Qualification program.
Author information
Authors and Affiliations
Contributions
SS wrote the main manuscript. SJ and YY discussed technical ideas and reviewed the manuscript.
Corresponding author
Ethics declarations
Ethical Approval
Not applicable.
Competing Interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Derivation of the NOSB Influence Function
Appendix: Derivation of the NOSB Influence Function
In this Appendix it is shown that the form of the influence function \(\omega\) can be obtained from the form of the CG smoothing function \(\Omega\). It is convenient to use the continuum expression for the CG bond forces:
where the integrals are over the entire body. Assume that the lattice forces are short-range, that is, \(d\ll R\). Also let \(\Omega\) be radially symmetric, that is, \(\Omega (\textbf{X},\textbf{x})\) depends only on \(|\textbf{X}-\textbf{x}|\). Further assume that \(\textbf{X}\) is in the interior of the body, that there are no external forces, and that the body and the deformation are both homogeneous. Define \(\textbf{P}=\textbf{Q}-\textbf{X}\). Then (104) becomes
where \(\textbf{F}_0\) is a function independent of \(\textbf{X}\). Since the body is in equilibrium,
Since d is small, terms in a Taylor expansion for \(\Omega\) near \(\textbf{X}\) higher than first order can be neglected for present purposes. From (105),
In view of (106),
where the stress tensor is given by
Now recall the peridynamic bond stress for a corrospondence material with influence function \(\omega\):
in which, under the present assumptions, the shape tensor is isotropic, that is \(\textbf{K}=K_0\textbf{1}\) for some scalar \(K_0\). Comparing (108) and (110), the conclusion is that
where \(K_0\) is omitted, since the normalization of \(\omega\) is handled within the correspondence model. Under the present assumptions, \(\omega\) depends only on \(|\textbf{q}-\textbf{x}|\), and (111) therefore simplifies to
where \(\textbf{e}\) is any fixed unit vector and, taking liberties with the notation, \(\omega (|\textbf{q}-\textbf{x}|)\) replaces \(\omega (\textbf{q},\textbf{x})\). \({\mathcal {B}}\) is a neighborhood of \(\mathbf{{0}}\) with radius R. A typical curve for \(\omega\) is shown in Fig. 20, in which tent-shaped smoothing functions are assumed.
Equation (112) provides the expression for the influence function in an NOSB peridynamic material model derived from the underlying smoothing functions that are used for coarse graining.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Silling, S.A., Jafarzadeh, S. & Yu, Y. Peridynamic Models for Random Media Found by Coarse Graining. J Peridyn Nonlocal Model (2024). https://doi.org/10.1007/s42102-024-00118-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42102-024-00118-y