EMI 2021 - A comparative review of peridynamics and phase-field models for engineering fracture mechanics

A comparative review of peridynamics and phase-field
models for engineering fracture mechanics
Patrick Diehl
Joint work with: Thomas Wick, Robert Lipton, and Mayank Tyagi
Center for Computation & Technology, Louisiana State University
patrickdiehl@lsu.edu
January 2021
Patrick Diehl (CCT/LSU) Review: PD and PF January 2021 1 / 59

Motivation
Popularity
Phase-field and Peridynamic models are widely adapted in crack and
fracture simulations
The origin perdynamic papers have 4352 citations on Google Scholar
The origin phase-field papers have 2803 citations on Google Scholar
Relatively young methods
Peridynamics originated in 20001
Phase-field originated in 19982
References:
1 Silling, Stewart A. ”Reformulation of elasticity theory for discontinuities and
long-range forces.” Journal of the Mechanics and Physics of Solids 48.1 (2000):
175-209.
2 Francfort, Gilles A., and J-J. Marigo. ”Revisiting brittle fracture as an energy
minimization problem.” Journal of the Mechanics and Physics of Solids 46.8
(1998): 1319-1342.

Review’s outline
Section 2: Overview of the Numerical models
Section 3: Macroscale View of Crack Propagation Physics using
Thermodynamics Constraints and Constitutive Relationships
Section 4: Validation against experimental data
4.1: First Sandia Fracture Challenge
4.2: Third Sandia Fracture Challenge
4.3: Comparison of both models against the same experimental data
Section 5: Comparison between peridynamics and phase-field fracture
models
5.1 Computational aspects
5.2 Advantages
5.3 Challenges
Preprint
Diehl, Patrick, et al. “A Comparative Review of Peridynamics and Phase-field Models
for Engineering Fracture Mechanics.” engrXiv, 31 Mar. 2021. Web.

Overview
1 Computational aspects
2 Comparison between peridynamics and phase-field fracture models
Advantages
Challenges
3 Conclusion and Outlook

Computational aspects

Definitions
In this section we focus on the computational aspects of both models from
a bird’s eye view and compare the computations on a very high-level.
To do so we define the quantity of a field which can be a vector field
f = {f1, . . . , fn | fi ∈ R3} or a scalar field f = {f1, . . . , fn | fi ∈ R}.

Peridynamic
For peridynamic we have a so-called one field model where one solves
for the displacement field u and
the peridynamic damage field d(u) is evaluated using the
displacement field.
The displacement field is solved with explicit or implicit time
integration [30, 38, 75, 106, 165].
However, the majority of PD simulations utilized bond-based models due
to the increased computational costs for the state-based models. Similarly,
the majority of simulations utilized explicit time integration due to their
lower computational costs.

Phase-field
For phase-field we have a two field model with
the displacement field u and
damage field φ.
Approaches
For staggered schemes and alternating
minimization [21, 23, 28, 29, 73, 102] the global system is decoupled,
first, one solves for the displacement field u and second, one solves
for the phase-field damage field φ independently. For the equation of
motion, implicit or explicit time integration schemes can be utilized.
For the monolithic scheme [60, 78, 82, 105, 109, 117, 138, 157] the
displacement field and the phase-field damage field are fully coupled
and solved simultaneously.

Comparison between peridynamics and phase-field
fracture models

Advantages
1 Crack initiation:
One unique selling point of both models is that there is no need of
any initial crack, e.g. prescribed defects, in the model and cracks and
fractures initiated over time. Another notable approach with the same
feature is eigen-erosion framework [123, 124, 143, 150].
2 Notion of damage in the model representation:
In most other models or computational techniques an additional
criteria, e.g. as in Linear Elastic Fracture Mechanics, is needed to
describe the growth of cracks. However, in peridynamic and
phase-field models the criteria for the crack growth is determined as a
part of the solution and no external criteria is needed.
3 Increasing complexity in multi-field fracture:
Both models were extended to multi-field fracture.
PD [15, 33, 34, 48, 56, 63, 66, 76, 85, 89, 110, 118, 119, 119, 121, 122, 133, 135,
144, 145, 151, 159, 162, 168, 169]
PF [26, 35, 55, 67, 81, 87, 101, 103, 107, 137, 153–156, 158, 161, 166, 167, 170]

Common challenges
Computational cost:
Both PD and PF approaches are computationally expensive. For
peridynamics it is the meshless discretization, which is computational
intensive similar to molecular dynamics (MD) and smoothed-particle
hydrodynamics (SPH). Many implementations [42, 44, 47, 91, 108, 125, 126]
For phase-field models the length scale parameter l0 tends to become
small, thus, requiring small mesh sizes for the finite element.
Therefore, the method gets computational expensive due to the large
number of FEM elements. Phase-field models could be accelerated using a
staggered schemes instead of a monolith scheme [92], GPU acceleration [171], and
the Message Passing Interface (MPI) based parallelization [69, 88], and matrix-free
geometric multigrid methods [78, 79].

Common challenges
Lack of detailed three-dimensional simulations:
Probably due to their computational expenses, only few
three-dimensional simulations using PF and PD are reviewed here.
PD [16, 17, 21, 27, 31, 61, 74, 77, 93, 110, 114, 120, 134]
PF [19–21, 37, 64, 69, 72, 87, 92, 100, 105, 116, 128, 152]
Extraction of crack tip/surfaces:
Since both models have a notion of damage, the so-called phase-field
crack parameter φ and the peridynamic damage parameter d, the
position of the crack tip/surface is not encoded in the model and
needs to be approximated. This phenomena is not limited to
phase-field [14] and peridynamic models, e.g. [2, 65], and relates to
any other method which does not have explicit crack representation in
the model. This could be a source of error for tracking the crack tip
and compare the crack tip velocity against experimental observations
in dynamic fracture simulations.

Common challenges
Lack of validation studies against available experimental data:
Validation against experimental data for peridynamics is summarized
in [46] and for phase-field models in [160, Section 2.12]. However, for
an accurate comparison of these two models the same experiment or
a set of experiments should be utilized to gain some insights of both
methods on the same problem.
Unavailability in the commercial codes:
Most simulations of PF and PD models use their implementations in
corresponding scientific code bases. At the time of writing of this
review, not many commercial codes implemented either one of the
models. LS-DYNA provides a bond-based peridynamics
implementation discretized with the discontinuous Galerkin
FEM [134].

Common challenges
Unavailability in the commercial codes:
Most simulations of PF and PD models use their implementations in
corresponding scientific code bases. At the time of writing of this
review, not many commercial codes implemented either one of the
models. LS-DYNA provides a bond-based peridynamics
implementation discretized with the discontinuous Galerkin FEM [63].
Crack nucleation:
Despite crack nucleation being contained in both models, a final un-
derstanding from a theoretical point of view has not yet been fully
achieved. However, simulations are available for PF [132, 305, 371]
and PD [372–374].

Common challenges
Incompressible hyper elastic material behavior:
At the time of completing this review, not many material models or
simulations for incompressible hyper elastic material behavior, were
available for PF [208, 209] and PD [103, 375, 376]. Note that
modeling of hyper elastic material behavior is challenging for any
numerical method since the constitutive material law must reflect
material behaviors such as a neo-Hookean [377] or Mooney–Rivlin
[378] solids.

Common challenges
Microscale view of crack propagation physics using molecular
dynamics (MD) simulations:
Seleson et al. [97] showed that peridynamics (PD) model can recover
the same dynamics as the MD model through appropriate selection of
length scale for smooth deformations. Ahadi and Melin [379]
investigated accuracy of PD in capturing features emerging from
atomistic simulation [380] through calibration of interparticle bond
strength and length scale parameters elastic plastic effects. In a
similar attempt to connect the phase-field method to MD, Patil et al.
[381] derived PFM parameters from the MD atomistic simulations
and showed that the theoretical energy release rate G and internal
length param

Specific challenges for Peridynamics
Application of boundary conditions:
As mentioned in [46] a major challenge within PD is the treatment of
boundary conditions in a non-local fashion [50, 62, 94, 95, 129]. One
approach is to couple local and non-local models to enforce boundary
conditions in the local region and have the non-local model in the
region where crack and fractures arise. Fore more details we refer to
the review on non-local coupling approaches [39].

Choice of discretization parameters:
As mentioned in [46] the choice of the nodal spacing and the horizon
results in diverse convergence scenarios [51, 146]. One challenge is to
find the proper ratio between the horizon and mesh size, since the
simulations are sensitive [43] with respect of these parameters. One
adjustment is to select the ratio such that the PD simulation matches
the dispersion curve obtained by the experiment [141]. Another
adjustment is to determine the horizon by Griffith’s brittle failure
criterion [45]. To determine the discretization parameters from
experimental data, the peridynamic formulation of the virtual field
method could be applied [40].

Ductile fracture:
As the time of writing this review, not many material models and
simulation for ductile fracture were available [12, 36, 164]. Note that
ductile tearing is challenging for any numerical method, due to the
choice of an appropriate ductile failure model. This failure model
needs to incorporate the failure of hydrostatic stress (or triaxiality of
stress) to predict ductile failure.
Crack nucleation:
Despite that crack modeling is contained in the model, more work is
needed to fully understand crack nucleation in PD, however, some
work has been done [90, 115, 142].

Asymptotically compatible quadrature methods: Another way to
control accuracy of peridynamic methods is through development of
asymptotically compatible quadrature methods for fracture as in
recent approach of [95]. Here the numerical scheme is designed to
recover linear elastic behavior away from the crack set asymptotically
as the horizon tends to zero.

Specific challenges for Phase-field
Crack path intersecting holes, obstacles, and boundaries:
Several issues were reported while obtaining crack paths in agreement
with LEFM for problems involving holes [160]. Another study [54]
concluded that judgement on if a crack arrests or the method simply
does not permit continuation across obstacles, requires expert
knowledge. In pressurized fractures, see e.g., [157], the fracture often
branches, which raises however the question whether this is physically
reasonable. Moreover, goal functional evaluations are sensitive to
boundary conditions and the domain size [69].
Fast crack propagation under dynamic loading:
For some fast crack propagation simulations, e.g. [1], the calculated
fracture velocity overestimated the fracture energy dissipation.
Composite/Concrete fracture: At the time of writing this review,
not many fracture/damage models for composites [13, 49, 127] and
concrete [57, 113, 136, 163] were available.

Specific challenges for Phase-field
Crack nucleation: Despite crack nucleation is contained in the
model, a final understanding from a theoretical point of view has not
yet been fully achieved [84].
Asymptotic computational understanding of the interaction of
regularization, model, and discretization parameters: In terms of
numerical and computational convergence analysis, a final
understanding has definitely not yet been achieved. Ingredients of
numerical analysis from image segmentation [22], phase-field in fluid
flow [58, 59] are available. Furthermore, as we described previously,
computational convergences analyses for phase-field fracture have
been undertaken [68, 154]. Such a rigorous numerical analysis for a
phase-field fracture model substantiated with numerical tests is
missing to date.

Conclusion and Outlook

Conclusion and Outlook
Conclusion
Both numerical methods can be retrieve a consistent microscale
physics of crack initiation and propagation
Despite both approaches being computationally challenging, the
advantages of capturing multiple fracture interactions with minimal
amount of phenomenological assumptions and closures make PD and
PF as good choice to understand engineering fracture mechanics.
Outlook
Both PD and PF must be evaluated against the same experimental
benchmark for reasonable comparison in a blind validation manner.
There is in general lack of comparative studies among these two
leading modeling approaches for fracture initiation and propagation
even for the same set of simple fracture experiments.

References I
[1] V. Agrawal and K. Dayal. Dependence of equilibrium griffith surface
energy on crack speed in phase-field models for fracture coupled to
elastodynamics. International Journal of Fracture, 207(2):243–249,
2017.
[2] A. Agwai, I. Guven, and E. Madenci. Predicting crack propagation
with peridynamics: a comparative study. International journal of
fracture, 171(1):65, 2011.
[3] R. Alessi, J.-J. Marigo, and S. Vidoli. Gradient damage models
coupled with plasticity: variational formulation and main properties.
Mechanics of Materials, 80:351–367, 2015.
[4] R. Alessi, S. Vidoli, and L. De Lorenzis. A phenomenological
approach to fatigue with a variational phase-field model: The
one-dimensional case. Engineering Fracture Mechanics, 190:53–73,
2018.

References II
[5] M. Ambati and L. De Lorenzis. Phase-field modeling of brittle and
ductile fracture in shells with isogeometric nurbs-based solid-shell
elements. Computer Methods in Applied Mechanics and
Engineering, 312:351–373, 2016.
[6] M. Ambati, T. Gerasimov, and L. De Lorenzis. Phase-field modeling
of ductile fracture. Computational Mechanics, 55(5):1017–1040,
2015.
[7] M. Ambati, R. Kruse, and L. De Lorenzis. A phase-field model for
ductile fracture at finite strains and its experimental verification.
Computational Mechanics, 57(1):149–167, 2016.
[8] G. Amendola, M. Fabrizio, and J. Golden. Thermomechanics of
damage and fatigue by a phase field model. Journal of Thermal
Stresses, 39(5):487–499, 2016.

References III
[9] F. Amiri, D. Millán, Y. Shen, T. Rabczuk, and M. Arroyo.
Phase-field modeling of fracture in linear thin shells. Theoretical and
Applied Fracture Mechanics, 69:102–109, 2014.
[10] P. Areias, T. Rabczuk, and M. Msekh. Phase-field analysis of
finite-strain plates and shells including element subdivision.
Computer Methods in Applied Mechanics and Engineering,
312:322–350, 2016.
[11] S. Basava, K. Mang, M. Walloth, T. Wick, and W. Wollner.
Adaptive and pressure-robust discretization of incompressible
pressure-driven phase-field fracture. submitted, SPP 1748 final
report, 2020.
[12] M. Behzadinasab. Peridynamic modeling of large deformation and
ductile fracture. PhD thesis, UT Austin, 2019.

References IV
[13] S. Biner and S. Y. Hu. Simulation of damage evolution in
composites: a phase-field model. Acta materialia, 57(7):2088–2097,
2009.
[14] J. Bleyer, C. Roux-Langlois, and J.-F. Molinari. Dynamic crack
propagation with a variational phase-field model: limiting speed,
crack branching and velocity-toughening mechanisms. International
Journal of Fracture, 204(1):79–100, 2017.
[15] F. Bobaru and M. Duangpanya. The peridynamic formulation for
transient heat conduction. International Journal of Heat and Mass
Transfer, 53(19-20):4047–4059, 2010.
[16] F. Bobaru, Y. D. Ha, and W. Hu. Damage progression from impact
in layered glass modeled with peridynamics. Central European
Journal of Engineering, 2(4):551–561, 2012.

References V
[17] F. Bobaru and S. A. Silling. Peridynamic 3d models of nanofiber
networks and carbon nanotube-reinforced composites. In AIP
Conference Proceedings, volume 712, pages 1565–1570. American
Institute of Physics, 2004.
[18] J. Boldrini, E. B. de Moraes, L. Chiarelli, F. Fumes, and
M. Bittencourt. A non-isothermal thermodynamically consistent
phase field framework for structural damage and fatigue. Computer
Methods in Applied Mechanics and Engineering, 312:395–427, 2016.
[19] M. J. Borden, T. J. Hughes, C. M. Landis, A. Anvari, and I. J. Lee.
A phase-field formulation for fracture in ductile materials: Finite
deformation balance law derivation, plastic degradation, and stress
triaxiality effects. Computer Methods in Applied Mechanics and
Engineering, 312:130–166, 2016.

References VI
[20] M. J. Borden, T. J. Hughes, C. M. Landis, and C. V. Verhoosel. A
higher-order phase-field model for brittle fracture: Formulation and
analysis within the isogeometric analysis framework. Computer
[21] M. J. Borden, C. V. Verhoosel, M. A. Scott, T. J. Hughes, and
C. M. Landis. A phase-field description of dynamic brittle fracture.
217:77–95, 2012.
[22] B. Bourdin. Image segmentation with a finite element method.
Mathematical Modelling and Numerical Analysis, 33(2):229–244,
1999.
[23] B. Bourdin. Numerical implementation of a variational formulation
of quasi-static brittle fracture. Interfaces Free Bound.,
9(3):411–430, 08 2007.

References VII
[24] B. Bourdin, G. Francfort, and J.-J. Marigo. The variational
approach to fracture. J. Elasticity, 91(1-3):1–148, 2008.
[25] B. Bourdin, C. J. Larsen, and C. L. Richardson. A time-discrete
model for dynamic fracture based on crack regularization.
International journal of fracture, 168(2):133–143, 2011.
[26] B. Bourdin, J.-J. Marigo, C. Maurini, and P. Sicsic. Morphogenesis
and propagation of complex cracks induced by thermal shocks.
Physical review letters, 112(1):014301, 2014.
[27] M. Breitenfeld. Quasi-static non-ordinary state-based peridynamics
for the modeling of 3D fracture. PhD thesis, University of Illinois at
Urbana-Champaign, 2014.
[28] M. K. Brun, T. Wick, I. Berre, J. M. Nordbotten, and F. A. Radu.
An iterative staggered scheme for phase field brittle fracture
propagation with stabilizing parameters. Computer Methods in
Applied Mechanics and Engineering, 361:112752, 2020.

References VIII
[29] S. Burke, C. Ortner, and E. Süli. An adaptive finite element
approximation of a variational model of brittle fracture. SIAM J.
Numer. Anal., 48(3):980–1012, 2010.
[30] V. A. Buryachenko, C. Wanji, and Y. Shengqi. Effective
thermoelastic properties of heterogeneous thermoperistatic bar of
random structure. International Journal for Multiscale
Computational Engineering, 13(1), 2015.
[31] M. Bussler, P. Diehl, D. Pflüger, S. Frey, F. Sadlo, T. Ertl, and
M. A. Schweitzer. Visualization of fracture progression in
peridynamics. Computers & Graphics, 67:45–57, 2017.
[32] M. Caputo and M. Fabrizio. Damage and fatigue described by a
fractional derivative model. Journal of Computational Physics,
293:400–408, 2015.

References IX
[33] Z. Chen and F. Bobaru. Peridynamic modeling of pitting corrosion
damage. Journal of the Mechanics and Physics of Solids,
78:352–381, 2015.
[34] Z. Chen and F. Bobaru. Selecting the kernel in a peridynamic
formulation: a study for transient heat diffusion. Computer Physics
Communications, 197:51–60, 2015.
[35] C. Chukwudozie, B. Bourdin, and K. Yoshioka. A variational
phase-field model for hydraulic fracturing in porous media. Computer
[36] J. Conradie, T. Becker, and D. Turner. Peridynamic approach to
predict ductile and mixed-mode failure. R&D Journal, 35:1–8, 2019.
[37] T. Dally and K. Weinberg. The phase-field approach as a tool for
experimental validations in fracture mechanics. Continuum
Mechanics and Thermodynamics, 29(4):947–956, 2017.

References X
[38] 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(9):1811–1842, 2006.
[39] M. D’Elia, X. Li, P. Seleson, X. Tian, and Y. Yu. A review of
local-to-nonlocal coupling methods in nonlocal diffusion and
nonlocal mechanics. arXiv preprint arXiv:1912.06668, 2019.
[40] R. Delorme, P. Diehl, I. Tabiai, L. L. Lebel, and M. Lévesque.
Extracting constitutive mechanical parameters in linear elasticity
using the virtual fields method within the ordinary state-based
peridynamic framework. Journal of Peridynamics and Nonlocal
Modeling, pages 1–25, 2020.
[41] M. Diehl, M. Wicke, P. Shanthraj, F. Roters, A. Brueckner-Foit, and
D. Raabe. Coupled crystal plasticity–phase field fracture simulation
study on damage evolution around a void: pore shape versus
crystallographic orientation. JOM, 69(5):872–878, 2017.

References XI
[42] P. Diehl. Implementierung eines Peridynamik-Verfahrens auf GPU.
Diplomarbeit, Institute of Parallel and Distributed Systems,
University of Stuttgart, 2012.
[43] P. Diehl, F. Franzelin, D. Pflüger, and G. C. Ganzenmüller.
Bond-based peridynamics: a quantitative study of mode i crack
opening. International Journal of Fracture, 201(2):157–170, 2016.
[44] P. Diehl, P. K. Jha, H. Kaiser, R. Lipton, and M. Levesque.
Implementation of peridynamics utilizing hpx–the c++ standard
library for parallelism and concurrency. arXiv preprint
arXiv:1806.06917, 2018.
[45] P. Diehl, R. Lipton, and M. Schweitzer. Numerical verification of a
bond-based softening peridynamic model for small displacements:
deducing material parameters from classical linear theory. Institut
für Numerische Simulation Preprint, (1630), 2016.

References XII
[46] P. Diehl, S. Prudhomme, and M. Lévesque. A review of benchmark
experiments for the validation of peridynamics models. Journal of
Peridynamics and Nonlocal Modeling, 1(1):14–35, 2019.
[47] P. Diehl and M. A. Schweitzer. Efficient neighbor search for particle
methods on gpus. In Meshfree Methods for Partial Differential
Equations VII, pages 81–95. Springer, 2015.
[48] C. Diyaroglu, S. Oterkus, E. Oterkus, E. Madenci, S. Han, and
Y. Hwang. Peridynamic wetness approach for moisture
concentration analysis in electronic packages. Microelectronics
Reliability, 70:103–111, 2017.
[49] D. H. Doan, T. Q. Bui, N. D. Duc, and K. Fushinobu. Hybrid phase
field simulation of dynamic crack propagation in functionally graded
glass-filled epoxy. Composites Part B: Engineering, 99:266–276,
2016.

References XIII
[50] Q. Du. Nonlocal calculus of variations and well-posedness of
peridynamics. In Handbook of peridynamic modeling, pages
101–124. Chapman and Hall/CRC, 2016.
[51] Q. Du and X. Tian. Robust discretization of nonlocal models related
to peridynamics. In Meshfree methods for partial differential
equations VII, pages 97–113. Springer, 2015.
[52] F. P. Duda, A. Ciarbonetti, P. J. Sánchez, and A. E. Huespe. A
phase-field/gradient damage model for brittle fracture in
elastic–plastic solids. International Journal of Plasticity, 65:269–296,
2015.
[53] F. P. Duda, A. Ciarbonetti, S. Toro, and A. E. Huespe. A
phase-field model for solute-assisted brittle fracture in elastic-plastic
solids. International Journal of Plasticity, 102:16–40, 2018.

References XIV
[54] A. Egger, U. Pillai, K. Agathos, E. Kakouris, E. Chatzi, I. A.
Aschroft, and S. P. Triantafyllou. Discrete and phase field methods
for linear elastic fracture mechanics: a comparative study and
state-of-the-art review. Applied Sciences, 9(12):2436, 2019.
[55] W. Ehlers and C. Luo. A phase-field approach embedded in the
theory of porous media for the description of dynamic hydraulic
fracturing. Computer Methods in Applied Mechanics and
Engineering, 315:348–368, 2017.
[56] H. Fan and S. Li. A peridynamics-sph modeling and simulation of
blast fragmentation of soil under buried explosive loads. Computer
methods in applied mechanics and engineering, 318:349–381, 2017.
[57] D.-C. Feng and J.-Y. Wu. Phase-field regularized cohesive zone
model (czm) and size effect of concrete. Engineering Fracture
Mechanics, 197:66–79, 2018.

References XV
[58] X. Feng and A. Prohl. Numerical analysis of the Allen-Cahn
equation and approximation for mean curvature flows. Numerische
Mathematik, 94:33–65, 2003.
[59] X. Feng and A. Prohl. Analysis of a Fully Discrete Finite Element
Method for the Phase Field Model and Approximation of Its Sharp
Interface Limits. Math. Comp., 73:541–567, 2004.
[60] T. Gerasimov and L. De Lorenzis. A line search assisted monolithic
approach for phase-field computing of brittle fracture. Computer
[61] W. Gerstle, N. Sakhavand, and S. Chapman. Peridynamic and
continuum models of reinforced concrete lap splice compared.
Fracture Mechanics of Concrete and Concrete Structures-Recent
Advances in Fracture Mechanics of Concrete, 2010.

References XVI
[62] X. Gu, E. Madenci, and Q. Zhang. Revisit of non-ordinary
state-based peridynamics. Engineering Fracture Mechanics,
190:31–52, 2018.
[63] X. Gu, Q. Zhang, and E. Madenci. Refined bond-based peridynamics
for thermal diffusion. Engineering Computations, 2019.
[64] O. Gültekin, H. Dal, and G. A. Holzapfel. A phase-field approach to
model fracture of arterial walls: theory and finite element analysis.
Computer methods in applied mechanics and engineering,
312:542–566, 2016.
[65] Y. D. Ha and F. Bobaru. Studies of dynamic crack propagation and
crack branching with peridynamics. International Journal of
Fracture, 162(1-2):229–244, 2010.

References XVII
[66] G. Hattori, J. Trevelyan, C. E. Augarde, W. M. Coombs, and A. C.
Aplin. Numerical simulation of fracking in shale rocks: current state
and future approaches. Archives of Computational Methods in
Engineering, 24(2):281–317, 2017.
[67] Y. Heider and B. Markert. A phase-field modeling approach of
hydraulic fracture in saturated porous media. Mechanics Research
Communications, 80:38–46, 2017.
[68] T. Heister, M. F. Wheeler, and T. Wick. A primal-dual active set
method and predictor-corrector mesh adaptivity for computing
fracture propagation using a phase-field approach. Computer
[69] T. Heister and T. Wick. Parallel solution, adaptivity, computational
convergence, and open-source code of 2d and 3d pressurized
phase-field fracture problems. PAMM, 18(1):e201800353, 2018.

References XVIII
[70] C. Hesch, A. Gil, R. Ortigosa, M. Dittmann, C. Bilgen, P. Betsch,
M. Franke, A. Janz, and K. Weinberg. A framework for polyconvex
large strain phase-field methods to fracture. Computer Methods in
Applied Mechanics and Engineering, 317:649–683, 2017.
[71] C. Hesch and K. Weinberg. Thermodynamically consistent
algorithms for a finite-deformation phase-field approach to fracture.
International Journal for Numerical Methods in Engineering,
99(12):906–924, 2014.
[72] M. Hofacker and C. Miehe. Continuum phase field modeling of
dynamic fracture: variational principles and staggered fe
implementation. International Journal of Fracture,
178(1-2):113–129, 2012.

References XIX
[73] M. Hofacker and C. Miehe. A phase field model of dynamic fracture:
Robust field updates for the analysis of complex crack patterns.
93(3):276–301, 2013.
[74] W. Hu, Y. Wang, J. Yu, C.-F. Yen, and F. Bobaru. Impact damage
on a thin glass plate with a thin polycarbonate backing.
International Journal of Impact Engineering, 62:152–165, 2013.
[75] D. Huang, G. Lu, and P. Qiao. An improved peridynamic approach
for quasi-static elastic deformation and brittle fracture analysis.
International Journal of Mechanical Sciences, 94:111–122, 2015.
[76] S. Jafarzadeh, Z. Chen, and F. Bobaru. Peridynamic modeling of
intergranular corrosion damage. Journal of The Electrochemical
Society, 165(7):C362–C374, 2018.

References XX
[77] S. Jafarzadeh, Z. Chen, J. Zhao, and F. Bobaru. Pitting, lacy
covers, and pit merger in stainless steel: 3d peridynamic models.
Corrosion Science, 150:17–31, 2019.
[78] D. Jodlbauer, U. Langer, and T. Wick. Matrix-free multigrid solvers
for phase-field fracture problems. Computer Methods in Applied
Mechanics and Engineering, 372:113431, 2020.
[79] D. Jodlbauer, U. Langer, and T. Wick. Parallel matrix-free
higher-order finite element solvers for phase-field fracture problems.
Mathematical and Computational Applications, 25(3):40, 2020.
[80] M. Klinsmann, D. Rosato, M. Kamlah, and R. M. McMeeking.
Modeling crack growth during li insertion in storage particles using a
fracture phase field approach. Journal of the Mechanics and Physics
of Solids, 92:313–344, 2016.

References XXI
[81] C. Kuhn and R. Müller. Phase field simulation of thermomechanical
fracture. In PAMM: Proceedings in Applied Mathematics and
Mechanics, volume 9, pages 191–192. Wiley Online Library, 2009.
[82] C. Kuhn and R. Müller. A continuum phase field model for fracture.
Engineering Fracture Mechanics, 77(18):3625–3634, 2010.
[83] C. Kuhn, T. Noll, and R. Müller. On phase field modeling of ductile
fracture. GAMM-Mitteilungen, 39(1):35–54, 2016.
[84] A. Kumar, B. Bourdin, G. A. Francfort, and O. Lopez-Pamies.
Revisiting nucleation in the phase-field approach to brittle fracture.
Journal of the Mechanics and Physics of Solids, 142:104027, 2020.
[85] X. Lai, B. Ren, H. Fan, S. Li, C. Wu, R. A. Regueiro, and L. Liu.
Peridynamics simulations of geomaterial fragmentation by impulse
loads. International Journal for Numerical and Analytical Methods
in Geomechanics, 39(12):1304–1330, 2015.

References XXII
[86] C. J. Larsen, C. Ortner, and E. Süli. Existence of solutions to a
regularized model of dynamic fracture. Mathematical Models and
Methods in Applied Sciences, 20(07):1021–1048, 2010.
[87] S. Lee, M. F. Wheeler, and T. Wick. Pressure and fluid-driven
fracture propagation in porous media using an adaptive finite
element phase field model. Computer Methods in Applied
Mechanics and Engineering, 305:111–132, 2016.
[88] T. Li, J.-J. Marigo, D. Guilbaud, and S. Potapov. Gradient damage
modeling of brittle fracture in an explicit dynamics context.
108(11):1381–1405, 2016.
[89] Y. Liao, L. Liu, Q. Liu, X. Lai, M. Assefa, and J. Liu. Peridynamic
simulation of transient heat conduction problems in functionally
gradient materials with cracks. Journal of Thermal Stresses,
40(12):1484–1501, 2017.

References XXIII
[90] D. J. Littlewood. A nonlocal approach to modeling crack nucleation
in aa 7075-t651. In ASME International Mechanical Engineering
Congress and Exposition, volume 54945, pages 567–576, 2011.
[91] D. J. Littlewood. Roadmap for peridynamic software
implementation. Technical Report 2015-9013, Sandia National
Laboratories, 2015.
[92] G. Liu, Q. Li, M. A. Msekh, and Z. Zuo. Abaqus implementation of
monolithic and staggered schemes for quasi-static and dynamic
fracture phase-field model. Computational Materials Science,
121:35–47, 2016.
[93] W. Liu and J.-W. Hong. Discretized peridynamics for brittle and
ductile solids. International journal for numerical methods in
engineering, 89(8):1028–1046, 2012.

References XXIV
[94] E. Madenci, M. Dorduncu, A. Barut, and N. Phan. A state-based
peridynamic analysis in a finite element framework. Engineering
Fracture Mechanics, 195:104–128, 2018.
[95] E. Madenci, M. Dorduncu, A. Barut, and N. Phan. Weak form of
peridynamics for nonlocal essential and natural boundary conditions.
337:598–631, 2018.
[96] K. Mang, T. Wick, and W. Wollner. A phase-field model for
fractures in nearly incompressible solids. Computational Mechanics,
65:61–78, 2020.
[97] E. Martínez-Pañeda, A. Golahmar, and C. F. Niordson. A phase field
formulation for hydrogen assisted cracking. Computer Methods in
Applied Mechanics and Engineering, 342:742–761, 2018.

References XXV
[98] S. May, J. Vignollet, and R. De Borst. A numerical assessment of
phase-field models for brittle and cohesive fracture: γ-convergence
and stress oscillations. European Journal of Mechanics-A/Solids,
52:72–84, 2015.
[99] A. Mesgarnejad, B. Bourdin, and M. Khonsari. A variational
approach to the fracture of brittle thin films subject to out-of-plane
loading. Journal of the Mechanics and Physics of Solids,
61(11):2360–2379, 2013.
[100] A. Mesgarnejad, B. Bourdin, and M. Khonsari. Validation
simulations for the variational approach to fracture. Computer

References XXVI
[101] C. Miehe, M. Hofacker, L.-M. Schänzel, and F. Aldakheel. Phase
field modeling of fracture in multi-physics problems. part ii. coupled
brittle-to-ductile failure criteria and crack propagation in
thermo-elastic–plastic solids. Computer Methods in Applied
[102] C. Miehe, M. Hofacker, and F. Welschinger. A phase field model for
rate-independent crack propagation: Robust algorithmic
implementation based on operator splits. Computer Methods in
Applied Mechanics and Engineering, 199(45-48):2765–2778, 2010.
[103] C. Miehe, L.-M. Schaenzel, and H. Ulmer. Phase field modeling of
fracture in multi-physics problems. part i. balance of crack surface
and failure criteria for brittle crack propagation in thermo-elastic
solids. Computer Methods in Applied Mechanics and Engineering,
294:449–485, 2015.

References XXVII
[104] C. Miehe and L.-M. Schänzel. Phase field modeling of fracture in
rubbery polymers. part i: Finite elasticity coupled with brittle failure.
Journal of the Mechanics and Physics of Solids, 65:93–113, 2014.
[105] C. Miehe, F. Welschinger, and M. Hofacker. Thermodynamically
consistent phase-field models of fracture: Variational principles and
multi-field fe implementations. International Journal for Numerical
Methods in Engineering, 83(10):1273–1311, 2010.
[106] Y. Mikata. Analytical solutions of peristatic and peridynamic
problems for a 1d infinite rod. International Journal of Solids and
Structures, 49(21):2887–2897, 2012.
[107] A. Mikelic, M. F. Wheeler, and T. Wick. A phase-field method for
propagating fluid-filled fractures coupled to a surrounding porous
medium. Multiscale Modeling & Simulation, 13(1):367–398, 2015.

References XXVIII
[108] F. Mossaiby, A. Shojaei, M. Zaccariotto, and U. Galvanetto. Opencl
implementation of a high performance 3d peridynamic model on
graphics accelerators. Computers & Mathematics with Applications,
74(8):1856–1870, 2017.
[109] M. A. Msekh, J. M. Sargado, M. Jamshidian, P. M. Areias, and
T. Rabczuk. Abaqus implementation of phase-field model for brittle
fracture. Computational Materials Science, 96:472–484, 2015.
[110] S. Nadimi, I. Miscovic, and J. McLennan. A 3d peridynamic
simulation of hydraulic fracture process in a heterogeneous medium.
Journal of Petroleum Science and Engineering, 145:444–452, 2016.
[111] T.-T. Nguyen, J. Bolivar, J. Réthoré, M.-C. Baietto, and
M. Fregonese. A phase field method for modeling stress corrosion
crack propagation in a nickel base alloy. International Journal of
Solids and Structures, 112:65–82, 2017.

References XXIX
[112] T. T. Nguyen, J. Yvonnet, Q.-Z. Zhu, M. Bornert, and C. Chateau.
A phase-field method for computational modeling of interfacial
damage interacting with crack propagation in realistic
microstructures obtained by microtomography. Computer Methods
in Applied Mechanics and Engineering, 312:567–595, 2016.
[113] V. P. Nguyen and J.-Y. Wu. Modeling dynamic fracture of solids
with a phase-field regularized cohesive zone model. Computer
Methods in Applied Mechanics and Engineering, 340:1000–1022,
2018.
[114] T. Ni, M. Zaccariotto, Q.-Z. Zhu, and U. Galvanetto. Coupling of
fem and ordinary state-based peridynamics for brittle failure analysis
in 3d. Mechanics of Advanced Materials and Structures, pages 1–16,
2019.
[115] S. Niazi, Z. Chen, and F. Bobaru. Crack nucleation in brittle and
quasi-brittle materials: A peridynamic analysis. 2020.

References XXX
[116] N. Noii and T. Wick. A phase-field description for pressurized and
non-isothermal propagating fractures. Computer Methods in Applied
Mechanics and Engineering, 351:860 – 890, 2019.
[117] T. Noll, C. Kuhn, and R. Müller. A monolithic solution scheme for a
phase field model of ductile fracture. PAMM, 17(1):75–78, 2017.
[118] S. Oterkus, E. Madenci, and A. Agwai. Fully coupled peridynamic
thermomechanics. Journal of the Mechanics and Physics of Solids,
64:1–23, 2014.
[119] S. Oterkus, E. Madenci, and A. Agwai. Peridynamic thermal
diffusion. Journal of Computational Physics, 265:71–96, 2014.
[120] H. Ouchi, A. Katiyar, J. T. Foster, M. M. Sharma, et al. A
peridynamics model for the propagation of hydraulic fractures in
naturally fractured reservoirs. SPE Journal, 22(04):1–082, 2017.

References XXXI
[121] H. Ouchi, A. Katiyar, J. York, J. T. Foster, and M. M. Sharma. A
fully coupled porous flow and geomechanics model for fluid driven
cracks: a peridynamics approach. Computational Mechanics,
55(3):561–576, 2015.
[122] R. Panchadhara, P. A. Gordon, and M. L. Parks. Modeling
propellant-based stimulation of a borehole with peridynamics.
International Journal of Rock Mechanics and Mining Sciences,
93:330–343, 2017.
[123] A. Pandolfi, B. Li, and M. Ortiz. Modeling fracture by material-point
erosion. International Journal of fracture, 184(1-2):3–16, 2013.
[124] A. Pandolfi and M. Ortiz. An eigenerosion approach to brittle
fracture. International Journal for Numerical Methods in
Engineering, 92(8):694–714, 2012.

References XXXII
[125] M. Parks, D. Littlewood, J. Mitchell, and S. Silling. Peridigm users’
guide. Technical Report SAND2012-7800, Sandia National
Laboratories, 2012.
[126] M. L. Parks, R. B. Lehoucq, S. J. Plimpton, and S. A. Silling.
Implementing peridynamics within a molecular dynamics code.
Computer Physics Communications, 179(11):777–783, 2008.
[127] R. Patil, B. Mishra, I. Singh, and T. Bui. A new multiscale phase
field method to simulate failure in composites. Advances in
Engineering Software, 126:9–33, 2018.
[128] K. Pham and K. Ravi-Chandar. The formation and growth of
echelon cracks in brittle materials. International Journal of Fracture,
206(2):229–244, 2017.
[129] S. Prudhomme and P. Diehl. On the treatment of boundary
conditions for bond-based peridynamic models. Computer Methods
in Applied Mechanics and Engineering, 372:113391, 2020.

References XXXIII
[130] M. Radszuweit and C. Kraus. Modeling and simulation of
non-isothermal rate-dependent damage processes in inhomogeneous
materials using the phase-field approach. Computational Mechanics,
60(1):163–179, 2017.
[131] A. Raina and C. Miehe. A phase-field model for fracture in
biological tissues. Biomechanics and modeling in mechanobiology,
15(3):479–496, 2016.
[132] J. Reinoso, M. Paggi, and C. Linder. Phase field modeling of brittle
fracture for enhanced assumed strain shells at large deformations:
formulation and finite element implementation. Computational
Mechanics, 59(6):981–1001, 2017.
[133] B. Ren, H. Fan, G. L. Bergel, R. A. Regueiro, X. Lai, and S. Li. A
peridynamics–sph coupling approach to simulate soil fragmentation
induced by shock waves. Computational Mechanics, 55(2):287–302,
2015.

References XXXIV
[134] B. Ren, C. Wu, and E. Askari. A 3d discontinuous galerkin finite
element method with the bond-based peridynamics model for
dynamic brittle failure analysis. International Journal of Impact
Engineering, 99:14–25, 2017.
[135] S. Rokkam, N. Phan, M. Gunzburger, S. Shanbhag, and K. Goel.
Meshless peridynamics method for modeling corrosion crack
propagation. In 6th International Conference on Crack Paths (CP
2018)(Verona, Italy). http://www. cp2018. unipr. it, 2018.
[136] D. Santillán, J. C. Mosquera, and L. Cueto-Felgueroso. Phase-field
model for brittle fracture. validation with experimental results and
extension to dam engineering problems. Engineering Fracture
Mechanics, 178:109–125, 2017.

References XXXV
[137] A. Schlüter, C. Kuhn, and R. Müller. Simulation of laser-induced
controlled fracturing utilizing a phase field model. Journal of
Computing and Information Science in Engineering, 17(2):021001,
2017.
[138] A. Schlüter, A. Willenbücher, C. Kuhn, and R. Müller. Phase field
approximation of dynamic brittle fracture. Computational
Mechanics, 54(5):1141–1161, 2014.
[139] M. Seiler, P. Hantschke, A. Brosius, and M. Kästner. A numerically
efficient phase-field model for fatigue fracture–1d analysis. PAMM,
18(1):e201800207, 2018.
[140] P. Shanthraj, B. Svendsen, L. Sharma, F. Roters, and D. Raabe.
Elasto-viscoplastic phase field modelling of anisotropic cleavage
fracture. Journal of the Mechanics and Physics of Solids, 99:19–34,
2017.

References XXXVI
[141] S. A. Silling. A coarsening method for linear peridynamics.
International Journal for Multiscale Computational Engineering,
9(6), 2011.
[142] S. A. Silling, O. Weckner, E. Askari, and F. Bobaru. Crack
nucleation in a peridynamic solid. International Journal of Fracture,
162(1-2):219–227, 2010.
[143] F. Stochino, A. Qinami, and M. Kaliske. Eigenerosion for static and
dynamic brittle fracture. Engineering Fracture Mechanics,
182:537–551, 2017.
[144] Y. Tao, X. Tian, and Q. Du. Nonlocal diffusion and peridynamic
models with neumann type constraints and their numerical
approximations. Applied Mathematics and Computation,
305:282–298, 2017.

References XXXVII
[145] X. Tian and Q. Du. Analysis and comparison of different
approximations to nonlocal diffusion and linear peridynamic
equations. SIAM Journal on Numerical Analysis, 51(6):3458–3482,
2013.
[146] X. Tian and Q. Du. Asymptotically compatible schemes and
applications to robust discretization of nonlocal models. SIAM
Journal on Numerical Analysis, 52(4):1641–1665, 2014.
[147] H. Ulmer, M. Hofacker, and C. Miehe. Phase field modeling of
fracture in plates and shells. PAMM, 12(1):171–172, 2012.
[148] C. V. Verhoosel and R. de Borst. A phase-field model for cohesive
fracture. International Journal for numerical methods in
Engineering, 96(1):43–62, 2013.
[149] J. Vignollet, S. May, R. De Borst, and C. V. Verhoosel. Phase-field
models for brittle and cohesive fracture. Meccanica,
49(11):2587–2601, 2014.

References XXXVIII
[150] K. Wang and W. Sun. A unified variational eigen-erosion framework
for interacting brittle fractures and compaction bands in
fluid-infiltrating porous media. Computer Methods in Applied
[151] Y. Wang, X. Zhou, and X. Xu. Numerical simulation of propagation
and coalescence of flaws in rock materials under compressive loads
using the extended non-ordinary state-based peridynamics.
Engineering Fracture Mechanics, 163:248–273, 2016.
[152] M. Wheeler, T. Wick, and W. Wollner. An augmented-Lagangrian
method for the phase-field approach for pressurized fractures. Comp.
Meth. Appl. Mech. Engrg., 271:69–85, 2014.

References XXXIX
[153] M. F. Wheeler, T. Wick, and S. Lee. IPACS: Integrated Phase-Field
Advanced Crack Propagation Simulator. An adaptive, parallel,
physics-based-discretization phase-field framework for fracture
propagation in porous media. Computer Methods in Applied
Mechanics and Engineering, 367:113124, 2020.
[154] T. Wick. Multiphysics Phase-Field Fracture: Modeling, Adaptive
Discretizations, and Solvers. De Gruyter, Berlin, Boston, 12 Oct.
2020.
[155] T. Wick. Coupling fluid-structure interaction with phase-field
fracture. Journal of Computational Physics, 327:67 – 96, 2016.

References XL
[156] T. Wick. Coupling fluid-structure interaction with phase-field
fracture: algorithmic details. In S. Frei, B. Holm, T. Richter,
T. Wick, and H. Yang, editors, Fluid-Structure Interaction:
Modeling, Adaptive Discretization and Solvers, Radon Series on
Computational and Applied Mathematics. Walter de Gruyter, Berlin,
2017.
[157] T. Wick. An error-oriented Newton/inexact augmented Lagrangian
approach for fully monolithic phase-field fracture propagation. SIAM
Journal on Scientific Computing, 39(4):B589–B617, 2017.
[158] Z. A. Wilson and C. M. Landis. Phase-field modeling of hydraulic
fracture. Journal of the Mechanics and Physics of Solids,
96:264–290, 2016.

References XLI
[159] F. Wu, S. Li, Q. Duan, and X. Li. Application of the method of
peridynamics to the simulation of hydraulic fracturing process. In
International Conference on Discrete Element Methods, pages
561–569. Springer, 2016.
[160] J.-Y. Wu, V. P. Nguyen, C. Thanh Nguyen, D. Sutula, S. Bordas,
and S. Sinaie. Phase field modelling of fracture. Advances in
Applied Mechanics, 53, 09 2019.
[161] T. Wu and L. De Lorenzis. A phase-field approach to fracture
coupled with diffusion. Computer Methods in Applied Mechanics
and Engineering, 312:196–223, 2016.
[162] F. Yan, X.-T. Feng, P.-Z. Pan, and S.-J. Li. A
continuous-discontinuous cellular automaton method for cracks
growth and coalescence in brittle material. Acta Mechanica Sinica,
30(1):73–83, 2014.

References XLII
[163] Z.-J. Yang, B.-B. Li, and J.-Y. Wu. X-ray computed tomography
images based phase-field modeling of mesoscopic failure in concrete.
Engineering Fracture Mechanics, 208:151–170, 2019.
[164] U. Yolum, A. Taştan, and M. A. Güler. A peridynamic model for
ductile fracture of moderately thick plates. Procedia Structural
Integrity, 2:3713–3720, 2016.
[165] M. Zaccariotto, F. Luongo, U. Galvanetto, et al. Examples of
applications of the peridynamic theory to the solution of static
equilibrium problems. The Aeronautical Journal,
119(1216):677–700, 2015.
[166] S. Zhou, X. Zhuang, and T. Rabczuk. A phase-field modeling
approach of fracture propagation in poroelastic media. Engineering
Geology, 240:189–203, 2018.

References XLIII
[167] S. Zhou, X. Zhuang, and T. Rabczuk. Phase-field modeling of
fluid-driven dynamic cracking in porous media. Computer Methods
in Applied Mechanics and Engineering, 350:169 – 198, 2019.
[168] X. Zhou and Y. Shou. Numerical simulation of failure of rock-like
material subjected to compressive loads using improved peridynamic
method. International Journal of Geomechanics, 17(3):04016086,
2016.
[169] X.-P. Zhou, X.-B. Gu, and Y.-T. Wang. Numerical simulations of
propagation, bifurcation and coalescence of cracks in rocks.
international journal of Rock Mechanics and Mining Sciences,
80:241–254, 2015.
[170] X. Zhuang, S. Zhou, M. Sheng, and G. Li. On the hydraulic
fracturing in naturally-layered porous media using the phase field
method. Engineering Geology, 266:105306, 2020.

References XLIV
[171] V. Ziaei-Rad and Y. Shen. Massive parallelization of the phase field
formulation for crack propagation with time adaptivity. Computer

EMI 2021 - A comparative review of peridynamics and phase-field models for engineering fracture mechanics

More Related Content

What's hot

Similar to EMI 2021 - A comparative review of peridynamics and phase-field models for engineering fracture mechanics

More from Patrick Diehl

Recently uploaded

EMI 2021 - A comparative review of peridynamics and phase-field models for engineering fracture mechanics