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A phase field model for cohesive fracture in micropolar continua

DOI:10.1016/j.cma.2020.113181

Authors:

Hyoung Suk Suh

Case Western Reserve University

Waiching Sun

Columbia University

Devin Thomas O'Connor

Sandia National Laboratories

While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted via variational or material-force-based phase field fracture models, these models often assume that the underlying elastic response of the material is non-polar and yet a length scale parameter must be introduced to enable the sharp cracks represented by a regularized implicit function. However, many materials with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and path-dependent regimes. This paper is intended to introduce a unified treatment that captures the size effect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase field fracture theory, along with the computational model and validation exercises, we explore the interacting size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstructures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress-strain and couple-stress-micro-rotation energy-conjugated pairs for a given regularization profile such that the macroscopic size-dependent responses of the micropolar continua are insensitive to the length scale parameter of the regularized interface. Then, we apply the variational principle to derive governing equations from the micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate the proper way to identify material parameters and the capacity of the new formulation to simulate complex crack patterns in the quasi-static regime.

Schematic of geometry and boundary conditions for the single edge notched tests.

…

Fracture patterns for single edge notched tests under different micropolar characteristic length l b .

…

Resultant micro-rotation field for single edge notched shear tests where ¯ u 1 = 0.0072 mm.

…

The force-displacement curves from the single edge notched tests with regularization length l c = 0.008 mm.

…

Observed crack patterns for asymmetric notched three-point bending tests: (a) ∆ ¯ u 2 = −2.0 × 10 −4 mm; (b) ∆ ¯ u 2 = −4.0 × 10 −4 mm; (c) ∆ ¯ u 2 = −8.0 × 10 −4 mm; (d) ∆ ¯ u 2 = −16.0 × 10 −4 mm.

…

Figures - uploaded by Waiching Sun

Content may be subject to copyright.

Content uploaded by Waiching Sun

Content may be subject to copyright.

Computer Methods in Applied Mechanics and Engineering manuscript No.

(will be inserted by the editor)

A phase ﬁeld model for cohesive fracture in micropolar continua1

Hyoung Suk Suh ·WaiChing Sun ·Devin O’Connor2

Received: April 24, 2020/ Accepted: date4

Abstract While crack nucleation and propagation in the brittle or quasi-brittle regime can be predicted5

via variational or material-force-based phase ﬁeld fracture models, these models often assume that the6

underlying elastic response of the material is non-polar and yet a length scale parameter must be intro-7

duced to enable the sharp cracks represented by a regularized implicit function. However, many materials8

with internal microstructures that contain surface tension, micro-cracks, micro-fracture, inclusion, cavity9

or those of particulate nature often exhibit size-dependent behaviors in both the path-independent and10

path-dependent regimes. This paper is intended to introduce a uniﬁed treatment that captures the size ef-11

fect of the materials in both elastic and damaged states. By introducing a cohesive micropolar phase ﬁeld12

fracture theory, along with the computational model and validation exercises, we explore the interacting13

size-dependent elastic deformation and fracture mechanisms exhibits in materials of complex microstruc-14

tures. To achieve this goal, we introduce the distinctive degradation functions of the force-stress-strain and15

couple-stress-micro-rotation energy-conjugated pairs for a given regularization proﬁle such that the macro-16

scopic size-dependent responses of the micropolar continua is insensitive to the length scale parameter of17

the regularized interface. Then, we apply the variational principle to derive governing equations from the18

micropolar stored energy and dissipative functionals. Numerical examples are introduced to demonstrate19

the proper way to identify material parameters and the capacity of the new formulation to simulate com-20

plex crack patterns in the quasi-static regime.21

Keywords micropolar damage, phase ﬁeld fracture, cohesive fracture, regularization length sensitivity22

1 Introduction23

The size effect and the corresponding length scale parameter associated with the phase ﬁeld fracture model24

for brittle or quasi-brittle materials have been a subject of intensive research in recent years [Francfort and25

Marigo,1998,Bourdin et al.,2008,de Borst and Verhoosel,2016,Wang and Sun,2017,Aldakheel et al.,26

2018,Wu and Nguyen,2018,Geelen et al.,2018,Bryant and Sun,2018,Choo and Sun,2018b,Qinami27

et al.,2019,Noii et al.,2020]. Due to the fact that the phase ﬁeld approach employ regularized (smoothed)28

implicit function to represent sharp interface, the physical interpretation of the length scale parameter29

(and in some cases, the lack thereof) has become a hotly debated topic among the computational fracture30

mechanics community. The lack of consensus on the deﬁnition of length scale has also been sometimes31

perceived as a weakness, especially when compared with the embedded discontinuity approaches such32

as XFEM or assumed strain models [Mo¨

es et al.,1999,Armero and Linder,2008,Wang and Sun,2018,33

2019a,b].34

Corresponding author: WaiChing Sun

Assistant Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, 614 SW Mudd, Mail

Code: 4709, New York, NY 10027 Tel.: 212-854-3143, Fax: 212-854-6267, E-mail: wsun@columbia.edu

2 Hyoung Suk Suh et al.

The early attempt to justify the introduction of the length scale parameter for phase ﬁeld fracture mod-35

els can be tracked back to the ﬁrst variational fracture model in Francfort and Marigo [1998] where the36

variational fracture model is expected to exhibit Γ−convergence and therefore may converge to the sharp37

interface model as the mesh size and the length scale parameter approaches zero. While this line of work38

(e.g., Bourdin et al. [2008] and May et al. [2015]) provides a theoretical justiﬁcation, in practice, the length39

scale parameter must still be sufﬁciently large compared to the mesh size in order to solve the phase ﬁeld40

governing equation. This could be problematic if the simulations are designed for boundary value prob-41

lems at the large scales (e.g., hydraulic fracture, faulting of geological formation) where the small mesh42

size, even concentrated at a local regions, become impractical.43

One strategy to overcome this issue is to decouple or eliminate the effect of the length scale parameters44

on the constitutive responses such that a relatively large length scale parameter can be used for numerical45

purposes without compromising the accuracy of the constitutive responses. To derive a phase ﬁeld fracture46

model that exhibits macroscopic responses independent or at least not sensitive to the length scale param-47

eters, Wu and Nguyen [2018] and Geelen et al. [2019] both introduce new crack surface density functionals48

and the corresponding degradation function derived from cohesive zone models such that the underlying49

traction-separation law is independent of the length scale parameter. Their simulations have shown that50

the resultant fracture patterns and the macroscopic constitutive responses are both insensitive to the length51

scale parameters in the sub-critical regime.52

Another strategy to circumvent the length scale issue is to determine the underlying relationship53

among the length scale parameters and other material parameters (e.g., Young’s modulus, tensile strength)54

that can be obtained from a speciﬁc set of experimental tests [Nguyen et al.,2016,Pham et al.,2017,Choo55

and Sun,2018a]. However, these previous works also show that the analytical expression of the length scale56

parameter may vary in according to the chosen inverse problems and hence the length scale parameter is57

likely only valid for backward calibration for a speciﬁc problem but cannot be used for general-purposed58

forward predictions. Furthermore, identifying the correct length scale parameter for a given inverse prob-59

lem does not imply that such a length scale parameter is sufﬁciently large to ensure the solvability of the60

discretized governing equation(s).61

Nevertheless, one key aspect that is often overlooked in the phase ﬁeld fracture modeling is that materials with62

internal structures that enables size effects on damage and fracture may likely exhibit size effect in the elastic regimes.63

Examples of these materials include concrete, composite, particulate materials as well as some metamate-64

rials [Dietsche et al.,1993,Bazant and Planas,1997,Trovalusci et al.,2014,Wang et al.,2016,Aldakheel,65

2020]. Since these materials exhibit large internal length scales compared to the length scale of damage or66

fracture, suitable size effect must be carefully incorporated in both the elastic and path-dependent regimes67

to capture the size-dependence properly. This paper is the ﬁrst attempt to formulate a new cohesive mi-68

cropolar phase ﬁeld fracture theory that leads to a physically justiﬁed/identiﬁable size-dependent effect69

for both the path-independent elastic responses and the path-dependent damage and fracture in a higher-70

order continuum undergoing inﬁnitesimal deformation. By extending the length-scale-parameter insensi-71

tive formulation that approximates cohesive-type of response to the micropolar materials, we introduce72

a third strategy where one may employ sufﬁciently large phase ﬁeld length scale parameters to address73

the numerical needs without comprising the correct size effect that should exhibit in the numerical simu-74

lations. In order words, the resultant model is the best of both worlds, one that beneﬁts from the physical75

justiﬁcation of having a consistent size effect in both the elastic and damage regimes and yet retains the76

convenience of the approach in Wu and Nguyen [2018], Geelen et al. [2019], and Wu et al. [2020].77

The rest of the paper is organized as follows. We ﬁrst brieﬂy summarize the theory of micropolar elas-78

ticity (Section 2), and introduce the strain energy split approach that enables one to explore the effects of the79

partitioned energy densities. We then extend the regularized length-scale-insensitive phase ﬁeld formula-80

tion to the micropolar material models such that the length scale parameter for the phase ﬁeld is insensitive81

to the macroscopic responses. This treatment then enables us to replicate the size effect characterized by82

the higher-order material parameters (e.g., bending and torsion stiffnesses) that can be experimentally83

sought. Furthermore, by enforcing the macroscopic responses not sensitive to the phase ﬁeld length scale84

parameter, it enables us to conduct simulations in a spatial domain without the size constraint imposed85

by the ratio between the phase ﬁeld length scale and mesh. For completeness, the details of the ﬁnite el-86

ement discretization and operator-split solution scheme are discussed. Numerical examples are given to87

Micropolar phase ﬁeld fracture 3

verify the implementation, provide evidences on how micropolarity affects the macroscopic behaviors for88

quasi-brittle materials, and showcase the applicability of the proposed models.89

As for notations and symbols, bold-faced and blackboard bold-faced letters denote tensors (including90

vectors which are rank-one tensors); the symbol ’·’ denotes a single contraction of adjacent indices of two91

tensors (e.g., a·b=aibior c·d=cijdjk); the symbol ‘:’ denotes a double contraction of adjacent indices92

of tensor of rank two or higher (e.g., C:ε=Cijk l εkl); the symbol ‘⊗’ denotes a juxtaposition of two vectors93

(e.g., a⊗b=aibj) or two symmetric second order tensors [e.g., (α⊗β)ijkl =αij βkl]. We also deﬁne identity94

tensors: I=δij,I=δik δjl , and ¯

I=δil δjk, where δij is the Kronecker delta. As for sign conventions, unless95

speciﬁed, the directions of the tensile stress and dilative pressure are considered as positive.96

2 Theory of micropolar elasticity97

In this section, we brieﬂy summarize the kinematic and constitutive relations of an isotropic micropolar98

elastic materials undergoing inﬁnitesimal deformation. In this case, the kinematics of micropolar materials99

is characterized by both the displacement ﬁeld and the micro-rotations. The resultant strain tensor is no100

longer symmetric due to the higher-order kinematics. We thus decompose the strain tensor into symmetric101

and skew-symmetric parts, which consequently enables us to split the stored energy density into three dif-102

ferent parts. This energy split approach opens the door for us to explore the effects of distinct degradation103

of partitioned energy conjugated pairs, which will be discussed later in this study.104

2.1 Kinematics105

Let us consider a micropolar elastic body B ⊂ R3with material points Pidentiﬁed by the position vec-

tors x∈ B that undergoes inﬁnitesimal deformation. As illustrated in Fig. 1, unlike the classical non-polar

(Boltzmann) approach, each material point experiences micro-rotation θ(x,t), in addition to the transla-

tional displacement u(x,t)at time t. The micro-rotation represents the local rotation of the material point

x, which is independent of the displacement ﬁeld. Consequently, the rotational part of the polar decom-

position of the displacement gradient (i.e., macro-rotation) is also independent of the micro-rotation. The

micropolar strain ¯εand micro-curvature ¯κcan be deﬁned as follows [Eringen,1966,Sachio et al.,1984,

Ehlers and Volk,1998,Eringen,2012,Atroshchenko and Bordas,2015,Grbˇ

ci´

c et al.,2018]:

¯ε=∇uT−3

E·θ, (1)

¯κ=∇θ, (2)

where 3

E=3

Eijk is the Levi-Civita permutation tensor. The deﬁnition of micropolar strain in Eq. (1) im-106

plies that the normal strains (i.e., diagonal entries of the micropolar strain tensor) that contributes to the107

stretching are equivalent to those in the classical approach, whereas the shear strains (i.e., off-diagonal108

entries of the micropolar strain tensor) are dependent on the micro-rotation. Since the micropolar strain109

tensor is non-symmetric, we therefore decompose the micropolar strain tensor into symmetric (¯εsym ) and110

skew-symmetric parts (¯εskew), i.e.,111

¯ε=1

2(∇u+∇uT)

| {z }

:=¯εsym

2(∇uT−∇u)−3

E·θ

| {z }

:=¯εskew

. (3)

Notice that ¯εsym is equivalent to the Boltzmann strain tensor in classical non-polar approach.112

4 Hyoung Suk Suh et al.

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Original

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conﬁguration

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Deformed

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conﬁguration

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✓

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Fig. 1: Kinematics of a micropolar continuum.

2.2 Constitutive model and strain energy split113

To ensure stable elastic responses, the micropolar strain energy must fulﬁll strong ellipticity condition. To114

fulﬁll this requirement, we consider a micropolar strain energy density ψe(¯ε, ¯κ)that takes a quadratic form:115

ψe(¯ε, ¯κ) = 1

2¯ε:C: ¯ε+1

2¯κ:D: ¯κ. (4)

Here, Cand Dare constitutive moduli that possess the major symmetry (i.e., C=Cijkl =Ckl ij , and116

D=Dijkl =Dkli j ):117

C=λ(I⊗I)+(µ+κ)I+µ¯

I;D=α(I⊗I)+β¯

I+γI, (5)

where λ,µ,κ,α,β, and γare the material constants. The strong ellipticity of the strain energy density118

deﬁned in Eq. (4) implies that the following inequalities must be hold [Diegele et al.,2004,Li and Lee,119

2009,Eringen,2012]:120

3λ+2µ+κ≥0 ; 2µ+κ≥0 ; κ≥0 ;

3α+β+γ≥0 ; γ+β≥0 ; γ−β≥0. (6)

These material constants, including the size-dependent ones are related to the following material parame-121

ters that have been individually identiﬁed via experiments [Yavari et al.,2002,Atroshchenko and Bordas,122

2015,Lakes and Drugan,2015]:123











E=(2µ+κ) (3λ+2µ+κ)

2λ+2µ+κYoung’s modulus,

G=2µ+κ

2shear modulus,

ν=λ

2λ+2µ+κPoisson’s ratio,

lt=sβ+γ

2µ+κcharacteristic length in torsion,

lb=rγ

2(2µ+κ)characteristic length in bending,

χ=β+γ

α+β+γpolar ratio,

N=rκ

2(µ+κ)coupling number, N∈[0, 1].

(7)

The relations in Eq. (7) indicates that the size-dependence of the elasticity responses are related to the124

higher-order kinematics and kinetic. Although identiﬁcation of the material parameters that characterize125

the size effect remains challenging as demonstrated in Bigoni and Drugan [2007] and Neff et al. [2010], these126

micropolar material parameters are obtainable through well-documented inverse problems or analytical127

solutions, at least for a subset of micropolar materials such as porous media [Bigoni and Drugan,2007].128

This unambiguity is helpful for practical purposes.129

Micropolar phase ﬁeld fracture 5

In Eq. (7), the characteristic lengths ltand lbimply the nonlocal nature of micropolar material by quan-130

tifying the range of couple stress through their relationship to the micro-curvature. The coupling number131

N, on the other hand, quantiﬁes the level of shear stress asymmetry that represents the degree of microp-132

olarity of the material, e.g., N=0 corresponds to the classical elasticity while N=1 corresponds to133

the couple-stress theory [Mindlin and Tiersten,1962,Mindlin,1963,McGregor and Wheel,2014]. In the134

remainder of this paper, unless speciﬁed, we set N=0.5 for micropolar continuum simulations.135

Since this study aims to develop a framework that explores the interaction between size-dependent136

micropolar elasticity and fracture mechanisms, we split the strain energy density into three different parts137

[based on the decomposition of the micropolar strain, cf. Eq. (3)]: (1) the Boltzmann part ψB

e(¯εsym ); (2) the138

micro-continuum coupling part ψC

e(¯εskew ); and (3) the pure micro-rotational part ψR

e(¯κ), i.e.,139

ψe(¯ε, ¯κ) = ψB

e(¯εsym ) + ψC

e(¯εskew ) + ψR

e(¯κ), (8)

where the partitioned strain energy densities can be written as:

ψB

e(¯εsym ) = 1

2hλ(¯εsym :I)2+(2µ+κ)¯εsym : ¯εsym i, (9)

ψC

e(¯εskew ) = 1

2κ¯εskew : ¯εskew, (10)

ψR

e(¯κ) = 1

2hα(¯κ:I)2+β¯κ: ¯κT+γ¯κ: ¯κi. (11)

The force stress ¯

σcan be found by taking partial derivative of the energy density with respect to the140

micropolar strain. By using Eq. (3), the force stress can also be partitioned into ¯

σBand ¯

σC, which are the141

results of the pure non-polar deformation and the micro-continuum coupling effects, respectively:142

σ=∂ψe

∂¯ε=∂

∂¯ε(ψB

e+ψC

e) = λ(¯εsym :I)I+(2µ+κ)¯εsym

| {z }

:=¯

σB

+κ¯εskew

| {z }

:=¯

σC

. (12)

Similarly, the couple stress ¯mRthat is caused by the pure micro-rotation can be obtained as follows:143

¯mR=∂ψe

∂¯κ=∂ψR

∂¯κ=α(¯κ:I)I+β¯κT+γ¯κ. (13)

Based on the split approach, notice that the partitioned energy densities in Eqs. (9)-(11) can be recovered144

by:145

ψB

e=1

2¯

σB: ¯εsym ;ψC

e=1

2¯

σC: ¯εskew ;ψR

e=1

2¯mR: ¯κ, (14)

where (¯

σ=¯

σB+¯

σC, ¯ε=¯εsym +¯εskew )and (¯mR, ¯κ)are the energy-conjugated pairs. This partition of en-146

ergy then provides a mean to introduce different degradation mechanisms for different kinematic modes.147

3 Phase ﬁeld model for damaged micropolar continua148

This section presents a variational phase ﬁeld framework to model cohesive fracture in micropolar materi-149

als. Our starting point is the energy split introduced in Section 2. We introduce distinct degradation for the150

Boltzmann, coupling and micro-rotational energy-conjugated pairs and derive, for the ﬁrst time, the ac-151

tion functional for the variational phase ﬁeld fracture framework for micropolar continua. The governing152

equations are then sought by seeking the stationary point, i.e., the Euler-Lagrange equation. To ensure that153

the size effect exhibited in the simulations are originated from the micropolar effect, we adopt the crack154

surface density functional originally proposed by Wu and Nguyen [2018] and Geelen et al. [2019] to elim-155

inate the sensitivity of the regularization length scale for the phase ﬁeld fracture in Boltzmann continua.156

Our 1D analysis in Section 3.5 and numerical results in Section 5.1 suggest that this same crack surface157

density functional may also eliminate the sensitivity of the regularization length scale parameter for the158

micropolar framework.159

6 Hyoung Suk Suh et al.

3.1 Phase ﬁeld approximation of cohesive fracture160

This study adopts a phase ﬁeld approach to represent cracks via an implicit function [Bourdin et al.,2008,161

Miehe et al.,2010a,Borden et al.,2012,Clayton and Knap,2015]. Let Γbe the discontinuous surface within162

a micropolar elastic body B. We approximate the fracture surface area AΓas AΓd, which is the volume163

integration of crack surface density Γd(d,∇d)over B,164

AΓ≈AΓd=ZB

Γd(d,∇d)dV, (15)

where dis the phase ﬁeld which varies from 0 in undamaged regions to 1 in completely damaged regions.165

In this study, we consider the following crack surface density functional, which is originally used to intro-166

duce elliptic regularization of the Mumford-Shah functional for image segmentation [Mumford and Shah,167

1989], i.e.,168

Γd(d,∇d)=1

c01

w(d) + lc(∇d·∇d);c0=4Z1

0qw(s)ds, (16)

where lcis the regularization length scale that governs the size of the diffusive crack zone, c0>0 is a169

normalization constant, and w(d)is one of the function that controls the shape of the regularized proﬁle of170

the phase ﬁeld [Clayton and Knap,2011,Mesgarnejad et al.,2015,Bleyer and Alessi,2018]. Γ−convergence171

requires that the sharp cracks can be recovered by reducing the length scale parameter lcto zero such that172

[Clayton and Knap,2015,Wu and Nguyen,2018],173

AΓ=lim

lc→0AΓd. (17)

The local dissipation function w(d)should be a monotonically increasing function of d, and we distinguish174

between the following two choices for this function:175

w(d) = (d2

d. (18)

The quadratic local dissipation w(d) = d2is the most widely used approach in simulating brittle fracture,176

and has become the standard in the phase ﬁeld approximation [Bourdin et al.,2008,Miehe et al.,2010a,b,177

Bourdin et al.,2011,Borden et al.,2012,Miehe et al.,2015a,c,Aldakheel et al.,2018,Choo and Sun,2018a,178

Bryant and Sun,2018,Chukwudozie et al.,2019]. The major disadvantage of the quadratic local dissipa-179

tion is that the damage evolution initiates as soon as the load is applied so that there is no pure elastic180

response. However, the linear model w(d) = d, combined with suitable degradation functions, describes181

the cohesive fracture that possesses a threshold energy that is independent of the regularization length lc

182

[Lorentz et al.,2012,Mesgarnejad et al.,2015,Lorentz,2017,Geelen et al.,2019]. Also, in this case, the ma-183

terial is characterized by an elastic phase until the stored energy density reaches the threshold value. Since184

this study aims to decouple the regularization length lcfor large-scale simulations, we adopt the linear185

dissipation function to take advantage of the aforementioned characteristics. The expression for the crack186

surface energy density in Eq. (16) then becomes:187

Γd(d,∇d)=3

8lc

d+3lc

8(∇d·∇d). (19)

3.2 Free energy functional188

It should be noted that the crack propagation within a body Bcorresponds to the creation of new free189

surfaces Γ. This implies that the rate of change of the internal energy should be equal to the rate of change190

of the surface energy that contributes to the crack growth. By assuming that this concept can be applied to191

the micropolar elastic material as well, the total potential energy Ψcan be deﬁned as follows [Bilgen and192

Weinberg,2019,Geelen et al.,2019]:193

Ψ=ZBψe(¯ε, ¯κ)dV +ZΓGcdΓ, (20)

Micropolar phase ﬁeld fracture 7

where Gcis critical energy release rate that quantiﬁes the resistance to cracking. Then, revisiting Eqs. (8)194

and (15), we approximate the functional by195

Ψ≈ZBψdV, (21)

with196

ψ=gB(d)ψB+

e(¯εsym ) + gC(d)ψC

e(¯εskew ) + gR(d)ψR

e(¯κ) + ψB−

e(¯εsym )

| {z }

:=ψbulk(¯εsym, ¯εskew , ¯κ,d)

+GcΓd(d,∇d), (22)

where ψbulk(¯εsym, ¯εskew, ¯κ,d)is the degrading elastic bulk energy, and gi(d)are the stiffness degradation197

functions for the corresponding ﬁctitious undamaged energy density parts ψi

e(i=B,C,R). Here, notice198

that we decompose ψB

e(¯εsym )into a positive and negative parts, and degrade only the positive part in order199

to avoid crack propagation under compression, i.e.,200

ψB

e=ψB+

e+ψB−

e. (23)

In this study, we adopt the spectral decomposition scheme of Miehe et al. [2010a], so that each part can be201

written as:202

ψB±

e=1

2hλh¯εsym :Ii2

±+(2µ+κ)¯εsym

±: ¯εsym

±i; ¯εsym

±=

∑

a=1D¯

εsym

aE±(na⊗na), (24)

where h•i±=(•±|•|)/2 is the Macaulay bracket operator, ¯

εsym

ais the principal Boltzmann strains, and203

naare the corresponding principal directions.204

In order to investigate the effects of each energy density part, one may assume that the partitioned205

strain energy densities can either be degraded (i∈D)or remain completely undamaged (i∈U), i.e.,206

gi(d) = (g(d)if i∈D

1 if i∈U;D∪U={B,C,R};D∩U=∅. (25)

Here, g(d)is a monotonically decreasing function that satisﬁes the following conditions [Pham and Marigo,207

2013]:208

g(0) = 1 ; g(1) = 0 ; g0(d)≤0 for d∈[0, 1], (26)

where the superposed prime denotes derivative with respect to d. Explicit form of this function is provided209

in Section 3.4. Notice that this general approach can be tailored to many different situations. For example,210

one may only degrade the pure Boltzmann part of the strain energy, i.e., D={B},U={C,R}:211

ψbulk(¯εsym, ¯εskew, ¯κ,d) = g(d)ψB+

e(¯εsym ) + ψC

e(¯εskew ) + ψR

e(¯κ) + ψB−

e(¯εsym ), (27)

or degrade the entire strain energy density, i.e., D={B,C,R},U=∅:212

ψbulk(¯εsym, ¯εskew, ¯κ,d) = g(d)hψB+

e(¯εsym ) + ψC

e(¯εskew ) + ψR

e(¯κ)i+ψB−

e(¯εsym ). (28)

3.3 Derivation of Euler-Lagrange equations via variational principle213

Let Vdenote an appropriate function space. Then, based on the fundamental lemma of calculus of varia-214

tions, the necessary condition for the energy functional Ψ:V→Rin Eq. (21) to have a local extremum at215

a point χ0∈Vis that,216

δψ

δχ(χ0) = 0, (29)

where ψis the energy density that is previously deﬁned in Eq. (22), χ:={u,θ,d}indicates the ﬁeld217

variables, and δ(•)/δχdenotes the functional derivative with respect to χ. Notice that Eq. (29) is the so-218

called Euler-Lagrange equations, which yield the governing partial differential equations to be solved.219

8 Hyoung Suk Suh et al.

The linear momentum balance equation can be recovered by seeking the stationary point where the220

functional derivative of ψwith respect to uvanishes. By assuming no body forces and by only considering221

the single derivative, we have,222

δψ

δu=∂ψ

∂u−∇· ∂ψ

∂∇u=0. (30)

By revisiting Eq. (22), we get:223

∂ψ

∂u=0, (31)

and by Eq. (12),224

∇· ∂ψ

∂∇u=∇·gB(d)∂ψB

∂∇u+gC(d)∂ψC

∂∇u=∇·hgB(d)¯

σB+gC(d)¯

σCi, (32)

since the decomposition of the micropolar strain in Eq. (3) yields the following:

∂ψB

∂∇u=∂ψB

∂¯εsym :∂¯εsym

∂∇u=∂ψB

∂¯εsym :1

2(I+¯

I)=∂ψB

∂¯εsym =¯

σB, (33)

∂ψC

∂∇u=∂ψC

∂¯εskew :∂¯εskew

∂∇u=∂ψC

∂¯εskew :1

2(¯

I−I)=∂ψC

∂¯εskew =¯

σC. (34)

Similarly, assuming no body couples, the balance of angular momentum can be obtained by searching the225

local extremum where the functional derivative of ψwith respect to the micro-rotation θvanishes, i.e.,226

δψ

δθ=∂ψ

∂θ−∇· ∂ψ

∂∇θ=0. (35)

The partial derivative of ψwith respect to θis:227

∂ψ

∂θ=gC(d)∂ψC

∂¯εskew :∂¯εskew

∂θ=−3

E:hgC(d)¯

σCi. (36)

By Eq. (13), the partial derivative of ψwith respect to ∇θbecomes:228

∇· ∂ψ

∂∇θ=∇·gR(d)∂ψR

∂¯κ=∇·hgR(d)¯mRi. (37)

The damage evolution equation (i.e., functional derivative of ψwith respect to the phase ﬁeld d) can also229

be recovered as follows:230

δψ

δd=∂ψ

∂d−∇· ∂ψ

∂∇d=0, (38)

where, by revisiting Eq. (19),231

∂ψ

∂d=g0(d)"∑

i∈D

ψi

e#+3Gc

8lc, (39)

and232

∇· ∂ψ

∂∇d=∇·3Gclc

4∇d. (40)

Finally, collecting the terms from Eqs. (30)-(40), we obtain the following coupled system of partial differ-

ential equations to be solved:

∇·hgB(d)¯

σB+gC(d)¯

σCi=0balance of linear momentum, (41)

∇·hgR(d)¯mRi+3

E:hgC(d)¯

σCi=0balance of angular momentum, (42)

g0(d)F+3

81−2l2

c∇2d=0 nondimensionalized damage evolution equation, (43)

where ∇2(•) = ∇·∇(•)is the Laplacian operator, and Fis the degrading nondimensionalized strain233

energy density:234

∑

i∈D

ψi

Gc/lc. (44)

Micropolar phase ﬁeld fracture 9

3.4 Crack irreversibility and degradation function235

As far as D6=∅, we prevent crack healing by following the treatment used in [Miehe et al.,2015b,Choo236

and Sun,2018a,Bryant and Sun,2018] which ensures the irreversibility constraint by enforcing the driving237

force to be non-negative. Although the stored energy density is split into three different parts, we simply238

introduce one distinct history function or driving force Hwhich is the pseudo-temporal maximum of the239

degrading nondimensionalized energy density. Inserting our deﬁnition into Eq. (43) gives:240

g0(d)H+3

81−2l2

c∇2d=0. (45)

Revisiting Section 3.1, the term cohesive denotes that the model should possess a threshold for the loading,241

where damage does not develop below this value. Therefore, we particularly restrict the crack growth to242

initiate above a threshold energy density ψcrit by using the following history function, in order to approxi-243

mate the cohesive response:244

H=max

τ∈[0,t]Fcrit +Fcrit F

Fcrit −1+;Fcrit =ψcrit

Gc/lc, (46)

where Fcrit is the nondimensionalized threshold energy. Note that Eq. (45) is the ﬁeld equation that is245

actually solved for the phase ﬁeld in this study.246

We complete our formulation by specifying the degradation function g(d). This study adopts a quasi-247

quadratic degradation function [Lorentz et al.,2011,2012,Geelen et al.,2019], which is a rational function248

of the phase ﬁeld d. The quasi-quadratic degradation function has an associated upper bound on the reg-249

ularization length lc, and is deﬁned as:250

g(d) = (1−d)2

(1−d)2+md (1+pd);lc≤3Gc

8(p+2)ψcrit

, (47)

where m≥1 is constant, and p≥1 is a shape parameter that controls the peak stress and the fracture251

responses.252

Recall that this study restricts the damage evolution to initiate above a threshold energy. In other words,253

below the threshold (i.e., F/Fcrit ≤1), the driving force and the phase ﬁeld should satisfy H=Fcrit and254

d=0, respectively. In this case, the damage evolution equation [Eq. (45)] becomes:255

g0(0)Fcrit +3

8=0. (48)

Since the degradation function [Eq. (47)] yields g0(0) = −m, we require256

m=3

8Fcrit

, (49)

in order to trivially satisfy Eq. (48). Again, the most common choice for the degradation function would257

be a simple quadratic function, i.e., g(d)=(1−d)2. A phase ﬁeld model that adopts the quadratic degra-258

dation function requires a particular critical energy ψcrit that depends on lcto satisfy Eq. (48). However,259

by using the quasi-quadratic degradation function in Eq. (47) with Eq. (49), it is noted that the threshold260

energy density ψcrit is no longer dependent on lc, since the degradation function itself is designed to auto-261

matically satisfy Eq. (48). It reveals that the elastic response (i.e., if stored energy is below the threshold) is262

regularization length independent. The regularization length insensitivity of the model, for the case where263

the stored energy exceeds the threshold, will be discussed in Section 3.5.264

10 Hyoung Suk Suh et al.

3.5 One-dimensional analysis on phase ﬁeld regularization length sensitivity265

In order to gain insights on the regularization length insensitive response, we consider a similiar 1D bound-266

ary value problem previously used in Wu and Nguyen [2018] and Geelen et al. [2019] for length scale267

analysis. Our major departure is that the material is now an analog to the micropolar material where an268

length scale dependent state variable (which replaces the micro-rotation due to the low-dimensional kine-269

matics) is introduced to replicate the size effect of the elasticity response. Consider a one-dimensional270

bar x∈[−L,L]subjected to a tensile loading on both ends. We assume that the length 2Lis sufﬁciently271

long enough so that the any possible boundary effects can be neglected. We deﬁne the strain measures as272

ε=du/dx, and ¯

κ=le(dθ/dx), where leis the length scale. Our goal here is to check whether the size-273

dependent responses in the damaged zone is sensitive to the regularization length scale for the phase ﬁeld274

lc. Note that this formulation does yield size-dependent responses in both elastic and damage zones, but275

the kinematics in 1D does not permit rotation. Hence, θand ¯

κno longer indicate micro-rotation and micro-276

curvature respectively. Thus, analogous to Eq. (8), we introduce a 1D size-dependent model of which the277

strain energy density takes a quadratic form as:278

ψe=1

2CBdu

dx2

2CCdu

dx−ledθ

dx2

2CRl2

edθ

dx2

, (50)

where CB>0, CC>0 and CR>0 are the material parameters. Notice that (1) this stored energy functional

does not admit non-trivial zero-energy mode provided that CB+CC>0 and CR>0 and (2) if the length

scale levanishes, Eq. (50) reduces to an energy functional for the classical Boltzmann continuum. In this

setting, the stress measures can be obtained as,

σ=∂ψe

∂¯

ε=CB¯

ε+CC(¯

ε−¯

κ), (51)

m=∂ψe

∂¯

κ=CR¯

κ−CC(¯

ε−¯

κ), (52)

where both ¯

σand ¯

mare le-dependent. Assuming all energy density parts can be degraded, the Lagrangian279

for the damaged state where ψe>ψcrit then becomes:280

ψ=g(d)ψe+3Gc

8lc"d+l2

cdd

dx2#, (53)

where g(d)is previously deﬁned in Eq. (47). The ﬁrst variation of Eq. (53) yields the following set of Euler-

Lagrange equations:

dx[g(d)¯

σ]=0, (54)

dx[g(d)¯

m]=0, (55)

g0(d)ψe

Gc/lc

81−2l2

d2d

dx2=0, (56)

where Eqs. (54) and (55) are the balance equations, and Eq. (56) is the nondimensionalized damage evo-281

lution equation. Following Geelen et al. [2019], we apply a speciﬁc amount of ¯

σat both ends while the282

boundaries remain ¯

m-free, such that Eq. (54) and Eq. (55) yields:283

g(d)¯

σ=¯

σ0;g(d)¯

m=0, (57)

where ¯

σ0is the responding stress on the boundary (x=±L). By the constitutive relationships in Eqs. (51)-284

(52), we can now express the strain measures ¯

εand ¯

κas,285

ε=CC−CR

CB(CC−CR)−CCCR

σ0

g(d);¯

κ=CC

CB(CC−CR)−CCCR

σ0

g(d). (58)

Micropolar phase ﬁeld fracture 11

Thus, the energy density functional in Eq. (50) can be rewritten as:286

ψe=¯

σ2

C∗g(d)2;C∗=2[CB(CC−CR)−CCCR]2

CB(CC−CR)2+CCCR(CC+CR). (59)

Substituting Eq. (57) and Eq. (59) into Eq. (56), we get,287

g0(d)"1

Gc/lc

σ2

C∗g(d)2#+3

81−2l2

d2d

dx2=0. (60)

By multiplying Eq. (60) with dd/dx, we can obtain the following differential equation:288

dx(3

8"d−l2

cdd

dx2#−1

Gc/lc

σ2

2C∗g(d))=0. (61)

Following Wu and Nguyen [2018] and Geelen et al. [2019], we focus on the damaged zone [−lz,lz]with289

lzL, where the outer edge of the zone is related to the parameter d∗, the maximum value of the damage290

across the bar (i.e., d∗=1 if the bar is fully broken). Then, by symmetry, at x=0 we have:291

d(d∗, 0) = d∗;dd

dx(d∗, 0) = 0, (62)

while the boundary conditions at the outer edge x=lzare given by,292

d(d∗,lz) = 0 ; dd

dx(d∗,lz) = 0. (63)

By introducing the parameter d∗, notice that our goal is to ﬁnd the expression for the responding stress ¯

σ0

293

as a function of the maximum damage d∗, or vice versa. Integration of Eq. (61), using boundary conditions294

in Eq. (63) admits the following:295

8"d−l2

cdd

dx2#=1

Gc/lc

σ2

2C∗g(d)−1−1. (64)

Applying the symmetry conditions [Eq. (62)], Eq. (64) becomes:296

σ2

2C∗=3Gc

8lc

d∗

g(d∗)−1−1. (65)

Substituting the expression of the degradation function in Eq. (47) and the expression for the degradation297

parameter min Eq. (49), we ﬁnally get:298

σ0=s2C∗ψcrit

(1−d∗)2

1+pd∗. (66)

Observe that the resultant stress ¯

σ0can be expressed in terms of d∗but independent of the phase ﬁeld299

regularization length lc. Eq. (66) highlights that the proposed model is capable of replicating the global300

response insensitive to regularization length lcfor the phase ﬁeld, while preserving the size effect intro-301

duced by the micropolar elasticity. This result is important, as this insensitivity to lcenables us to simulate302

cohesive fracture in large spatial domain composed of micropolar materials. Extending this analysis for 2D303

and 3D cases is out of the scope of this study but will be considered in the future.304

Remark 1 Previous works on phase ﬁeld and gradient damage models for cohesive fracture in Cauchy305

continuum, such as Cazes et al. [2010], Lorentz et al. [2012], Wu [2018], have established a connection306

between the cohesive zone models and the phase ﬁeld and gradient damage models that represent cracks307

via implicit function. In principle, it is possible that similar connection can be established between the308

micropolar phase ﬁeld model presented in this paper and the established micropolar cohesive zone models309

such as Larsson and Zhang [2007], Zhang et al. [2007], Hirschberger et al. [2008]. Such an endeavor is310

obviously out of the scope of this study due to the extensive length, but will be considered in future studies.311

12 Hyoung Suk Suh et al.

4 Finite element implementation312

In this section, we describe the ﬁnite element discretization, followed by the solution strategy to solve the313

system of nonlinear equation incrementally. Starting from the strong form, we follow the standard proce-314

dure to recover the variational form while employing the Taylor-Hood ﬁnite element space for the displace-315

ment and micro-rotation ﬁelds, and standard linear interpolation for the phase ﬁeld. This ﬁnite element316

space is chosen to match the design of the operator-split algorithm. The displacement and micro-rotation317

are updated in a monolithic manner, where we use Taylor-Hood element such that the displacement ﬁeld is318

interpolated by quadratic polynomials and the micro-rotation is interpolated by linear polynomials. Mean-319

while, the phase ﬁeld is also interpolated by linear function to ensure the efﬁciency of the staggered solver320

that updates the phase ﬁeld while holding the displacement and micro-rotation ﬁxed. The operator-split321

solution scheme (i.e., staggered scheme) that successively updates the ﬁeld variables is described in Section322

4.2.323

4.1 Galerkin form324

We derive the weak form and introduce the ﬁnite dimensional space to introduce the numerical scheme325

for the boundary value problems described in Eqs. (41)-(42) and Eq. (45). We consider a micropolar elastic326

domain Bwith boundary ∂Bcomposed of Dirichlet boundaries (displacement ∂Buand micro-rotation ∂Bθ)327

and Neumann boundaries (traction ∂Btσand moment ∂Btm) satisfying,328

∂B=∂Bu∪∂Btσ=∂Bθ∪∂Btm;∅=∂Bu∩∂Btσ=∂Bθ∩∂Btm. (67)

The prescribed boundary conditions can be speciﬁed as:











u=ˆuon ∂Bu,

θ=ˆ

θon ∂Bθ,

hgB(d)¯

σB+gC(d)¯

σCiT·n=ˆ

tσon ∂Btσ,

hgR(d)¯mRiT·n=ˆ

tmon ∂Btm,

∇d·n=0 on ∂B,

(68)

where nis the outward-oriented unit normal on the boundary surface ∂B; ˆu,ˆ

θ,ˆ

tσ,ˆ

tmare the prescribed329

displacement, micro-rotation, traction and moment, respectively; and the degradation functions gi(d)(i=330

B,C,R) are previously deﬁned in Eq. (25). For the model closure, the initial conditions are imposed as,331

u=u0;θ=θ0, (69)

at t=0.332

We deﬁne the trial spaces Vu,Vθ, and Vdfor the solution variables:

Vu=nu:B → R3|u∈[H1(B)]3,u|∂Bu=ˆuo, (70)

Vθ=nθ:B → R3|θ∈[H1(B)]3,θ|∂Bθ=ˆ

θo, (71)

Vd=nd:B → R|d∈H1(B)o, (72)

where H1denotes the Sobolev space of order 1. Notice that this study adopts Taylor-Hood ﬁnite ele-

ment (i.e., quadratic interpolation for displacement and linear for micro-rotation) following Verhoosel and

de Borst [2013], which showed that the cohesive fracture model exhibits stress oscillation when equal order

polynomials are used for the solution ﬁeld, while the discretization with high order interpolation function

Micropolar phase ﬁeld fracture 13

for the displacement and ﬁrst order functions for the auxiliary ﬁeld and the phase ﬁeld seems to elimi-

nate this oscillation. Similarly, the corresponding admissible spaces for Eqs. (70)-(72) with homogeneous

boundary conditions are deﬁned as,

Vη=nη:B → R3|η∈[H1(B)]3,η|∂Bu=0o, (73)

Vξ=nξ:B → R3|ξ∈[H1(B)]3,ξ|∂Bθ=0o, (74)

Vζ=nζ:B → R|ζ∈H1(B)o. (75)

Applying the standard weighted residual procedure, the weak statements for Eqs. (41)-(42) and Eq. (45) is333

to: ﬁnd {u,θ,d}∈Vu×Vθ×Vdsuch that for all {η,ξ,ζ}∈Vη×Vξ×Vζ,334

Gu(u,θ,d,η) = Gθ(u,θ,d,ξ) = Gd(u,θ,d,ζ) = 0. (76)

Here, Gu→Ris the weak statement of the balance of linear momentum:335

Gu=ZB∇η:hgB(d)¯

σB+gC(d)¯

σCidV −Z∂Btσ

η·ˆ

tσdA =0, (77)

Gθ→Ris the weak statement of the balance of angular momentum:336

Gθ=ZB∇ξ:gR(d)¯mRdV −ZBξ·3

E:hgC(d)¯

σCidV −Z∂Btm

ξ·ˆ

tmdA =0, (78)

and Gd→Ris the weak statement of the damage evolution equation:337

Gd=ZBζ·g0(d)HdV +3

8ZBζ+2l2

c(∇ζ·∇d)dV=0, (79)

where Hand g(d)are previously deﬁned in Eqs. (46) and (47), respectively.338

4.2 Operator-split solution scheme339

As previous studies on the phase ﬁeld model showed that the operator splitting (i.e., staggered scheme)340

may potentially be more robust compared to the monolithic approach [Miehe et al.,2010a,Heister et al.,341

2015,Teichtmeister et al.,2017], this study adopts the solution procedure based on the operator-split342

scheme to successively update three ﬁeld variables {u,θ,d}. In this operator-split setting, the damage ﬁeld343

is updated ﬁrst while the displacement and micro-rotation ﬁelds are held ﬁxed. A new damage ﬁeld dn+1

344

is obtained iteratively once the algorithm converges within a predeﬁned tolerance. Then, the linear solver345

holds the damage ﬁeld ﬁxed and advances the displacement and the micro-rotation ﬁelds {un+1,θn+1}. The346

schematic of the solution strategy can be summarized as follows:347





θn

dn

R(d)=0

−−−−→ 



θn

dn+1



| {z }

Iterative solver

Linear solver

z }| {

R(u,θ)=0

−−−−−→ 



un+1

θn+1

dn+1

, (80)

where R(u,θ)and R(d)are the residuals that are consistent with Eqs. (77)-(79):

R(u,θ):









ZB∇η:hgB(dn+1)¯

σB

n+1 +gC(dn+1)¯

σC

n+1idV −Z∂Btσ

η·ˆ

tσn+1 dA,

ZB∇ξ:gR(dn+1)¯mR

n+1 dV −ZBξ·3

E:hgC(dn+1)¯

σC

n+1idV −Z∂Btm

ξ·ˆ

tmn+1 dA

(81)

R(d):ZBζ·g0(dn+1)HndV +3

8ZBζ+2l2

c(∇ζ·∇dn+1)dV. (82)

14 Hyoung Suk Suh et al.

It should be noticed that one may choose other strategies to solve the same system of equations, however,348

the exploration of different schemes are out of the scope of this study.349

The implementation of the numerical models including the ﬁnite element discretization and the operator-350

split solution scheme rely on the ﬁnite element package FEniCS [Logg and Wells,2010,Logg et al.,2012a,b,351

Alnæs et al.,2015] with PETSc scientiﬁc computation toolkit [Abhyankar et al.,2018]. The scripts devel-352

oped for this study are open-sourced (available at https://github.com/hyoungsuksuh/micropolar_353

phasefield), in order to aid third-party veriﬁcation and validation [Suh and Sun,2019].354

5 Numerical examples355

This section presents numerical examples to showcase the applicability of the proposed phase ﬁeld model356

for damaged micropolar elastic material. For simplicity, we limit our attention to two-dimensional sim-357

ulations in this section. Based on the 2D setting, the kinematic state of the micropolar elastic body can358

be described by two in-plane displacements u= [u1,u2]Tand one out-of-plane micro-rotation angle θ3.359

Since the material elasticity now only depends on bending characteristic length, we now require only four360

engineering material parameters (e.g., E,ν,N, and lb).361

The ﬁrst example serves as a veriﬁcation test that highlights the regularization length insensitive re-362

sponse of the phase ﬁeld model with quasi-quadratic degradation function in Eq. (47). We then investigate363

the effect of scale-dependent elasticity on the crack patterns, by simulating asymmetric notched three-point364

bending tests with different coupling numbers Nand single edge notched tests with different characteristic365

lengths lb, respectively. Finally, we exhibit the applicability of the proposed energy split scheme by con-366

sidering different degradation functions on the partitioned energy densities. All the numerical simulations367

rely on meshes that are sufﬁciently reﬁned to properly capture the damage ﬁeld around crack surfaces. Un-368

less speciﬁed, we especially adopt the element size of he≈lc/10 around the potential crack propagation369

trajectory.370

5.1 Veriﬁcation exercise: the trapezoid problem371

We ﬁrst examine a problem proposed by Lorentz et al. [2012] that has a trapezoidal-shaped symmetrical372

domain with an initial notch. Since we prescribe the displacement ¯uin order to consider pure Mode I373

loading, as illustrated in Fig. 2, this speciﬁc geometry helps us to avoid crack kinking and to facilitate374

straight crack propagation. The aforementioned characteristics of the trapezoid problem makes it suitable375

for verifying the regularization length insensitive response of the cohesive phase ﬁeld model with quasi-376

quadratic degradation. This example thus performs a parametric study for three different regularization377

lengths (lc=7.5, 15, and 30 mm, as depicted in Fig. 2), with two different shape parameters p=2.5 and378

10.379

The material is assumed to be similar to the concrete studied in Lorentz et al. [2012]. The material380

parameters for this example are chosen as follows: Young’s modulus E=30 GPa, Poisson’s ratio ν=0.2,381

critical energy release rate Gc=0.1 N/mm, and threshold energy density ψcrit =0.1 kJ/m3. We also set382

coupling number N=0 and bending characteristic length lb=0 mm in order to avoid all the micropolar383

effects, and at the same time damage only the pure Boltzmann part, i.e., D={B},U={C,R}. We prescribe384

∆¯

u2=0.5 ×10−3mm on the left boundaries, while all other boundaries are maintained traction-free during385

the simulations.386

Fig. 3shows the force–displacement curves for the trapezoid problem, corroborated by other numerical387

observations [Lorentz et al.,2012,Geelen et al.,2019]. The results indicate that the shape parameter pis able388

to inﬂuence the peak force and the overall global force-displacement responses. In fact, the quasi-quadratic389

degradation function enables us to not only tailor the threshold for the elastic region by controlling ψcrit,390

but also tune the peak stress by varying shape parameter p. As pointed out in Geelen et al. [2019], higher391

value of ptend to signiﬁcantly elongate the length of the fracture process zone, such that the stresses in the392

process zone can effectively be smeared over a large distance. As a result, we can observe that an increasing393

value of pyields the global force-displacement response where the effect of using different length scale394

parameters lcbecomes negligible, as previously reported in Lorentz et al. [2012], Wu and Nguyen [2018],395

Geelen et al. [2019], Wu et al. [2020].396

Micropolar phase ﬁeld fracture 15

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¯

<latexit sha1_base64="v7ukw/+N0TL+pLJ5nU23m+L/olk=">AAAB9HicbZBLSwMxFIXv+Kz1VXXpJlgEN5aZKuiyIIjLCvYBnaFk0kwbmknGJFMow4D/wo0LRdz6Y9z5b0wfC209EPg454bcnDDhTBvX/XZWVtfWNzYLW8Xtnd29/dLBYVPLVBHaIJJL1Q6xppwJ2jDMcNpOFMVxyGkrHN5M8taIKs2keDDjhAYx7gsWMYKNtYJzP8Qq88MoS/O8Wyq7FXcqtAzeHMowV71b+vJ7kqQxFYZwrHXHcxMTZFgZRjjNi36qaYLJEPdpx6LAMdVBNl06R6fW6aFIKnuEQVP3940Mx1qP49BOxtgM9GI2Mf/LOqmJroOMiSQ1VJDZQ1HKkZFo0gDqMUWJ4WMLmChmd0VkgBUmxvZUtCV4i19ehma14l1UqveX5drt06yOAhzDCZyBB1dQgzuoQwMIPMIzvMKbM3JenHfnYza64swrPII/cj5/ADD9kss=</latexit>

3000 mm

<latexit sha1_base64="kOl8VKNEqsIAl8+5n2sjOJ41X3o=">AAAB+HicbZBLSwMxFIUz9VXro6Mu3QSL4KrMtIIuC4K4rGAf0A4lk2ba0CQzJHfEOhT8H25cKOLWn+LOf2P6WGjrgcDHOTfk5oSJ4AY879vJra1vbG7ltws7u3v7RffgsGniVFPWoLGIdTskhgmuWAM4CNZONCMyFKwVjq6meeueacNjdQfjhAWSDBSPOCVgrZ5brHqeh7vAHiDDUk56bskrezPhVfAXUEIL1XvuV7cf01QyBVQQYzq+l0CQEQ2cCjYpdFPDEkJHZMA6FhWRzATZbPEJPrVOH0extkcBnrm/b2REGjOWoZ2UBIZmOZua/2WdFKLLIOMqSYEpOn8oSgWGGE9bwH2uGQUxtkCo5nZXTIdEEwq2q4ItwV/+8io0K2W/Wq7cnpdq10/zOvLoGJ2gM+SjC1RDN6iOGoiiFD2jV/TmPDovzrvzMR/NOYsKj9AfOZ8/eDySvw==</latexit>

4000 mm

<latexit sha1_base64="b95IjaizoYvEie+z31ruj4twd3I=">AAAB+HicbZBLSwMxFIUz9VXro6Mu3QSL4KrM1IIuC4K4rGAf0A4lk2ba0CQzJHfEOhT8H25cKOLWn+LOf2P6WGjrgcDHOTfk5oSJ4AY879vJra1vbG7ltws7u3v7RffgsGniVFPWoLGIdTskhgmuWAM4CNZONCMyFKwVjq6meeueacNjdQfjhAWSDBSPOCVgrZ5brHqeh7vAHiDDUk56bskrezPhVfAXUEIL1XvuV7cf01QyBVQQYzq+l0CQEQ2cCjYpdFPDEkJHZMA6FhWRzATZbPEJPrVOH0extkcBnrm/b2REGjOWoZ2UBIZmOZua/2WdFKLLIOMqSYEpOn8oSgWGGE9bwH2uGQUxtkCo5nZXTIdEEwq2q4ItwV/+8io0K2X/vFy5rZZq10/zOvLoGJ2gM+SjC1RDN6iOGoiiFD2jV/TmPDovzrvzMR/NOYsKj9AfOZ8/ec6SwA==</latexit>

45

<latexit sha1_base64="AjViA0g8zIzQWbyh2rilnkgPQSw=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKezGiB4DgniMYB6YrGF20kmGzM4uM7NCWAJ+hBcPinj1b7z5N04eB00saCiquunuCmLBtXHdbyezsrq2vpHdzG1t7+zu5fcP6jpKFMMai0SkmgHVKLjEmuFGYDNWSMNAYCMYXk38xiMqzSN5Z0Yx+iHtS97jjBor3ZfPH9I244qNO/mCW3SnIMvEm5MCzFHt5L/a3YglIUrDBNW65bmx8VOqDGcCx7l2ojGmbEj72LJU0hC1n04vHpMTq3RJL1K2pCFT9fdESkOtR2FgO0NqBnrRm4j/ea3E9C79lMs4MSjZbFEvEcREZPI+6XKFzIiRJZQpbm8lbEAVZcaGlLMheIsvL5N6qeidFUu35ULl+mkWRxaO4BhOwYMLqMANVKEGDCQ8wyu8Odp5cd6dj1lrxplHeAh/4Hz+AF1KkSI=</latexit>

lc=7.5 mm

<latexit sha1_base64="lY35BAAp4zqI9NsgIOrOd8yBw2g=">AAAB/XicbZBLSwMxFIUzPmt9jY+dm2ARXA0zVakboSCIywr2Ae0wZNK0DU1mhuSOWIeiP8WNC0Xc+j/c+W9MHwttPRD4OOeG3JwwEVyD635bC4tLyyurubX8+sbm1ra9s1vTcaooq9JYxKoREs0Ej1gVOAjWSBQjMhSsHvYvR3n9jinN4+gWBgnzJelGvMMpAWMF9r4IKL7AJecMt4DdQ4alHAZ2wXXcsfA8eFMooKkqgf3Vasc0lSwCKojWTc9NwM+IAk4FG+ZbqWYJoX3SZU2DEZFM+9l4+yE+Mk4bd2JlTgR47P6+kRGp9UCGZlIS6OnZbGT+lzVT6Jz7GY+SFFhEJw91UoEhxqMqcJsrRkEMDBCquNkV0x5RhIIpLG9K8Ga/PA+1ouOdOMWb00L56mlSRw4doEN0jDxUQmV0jSqoiih6QM/oFb1Zj9aL9W59TEYXrGmFe+iPrM8flWiUcw==</latexit>

lc= 15 mm

<latexit sha1_base64="b0vC4CCt4ellGZK9sms66BQlk38=">AAAB/HicbZBLSwMxFIUzPmt9jXbpJlgEV2WmKroRCoK4rGAf0A5DJs20ocnMkNwRh6HiP3HjQhG3/hB3/hvTx0JbDwQ+zrkhNydIBNfgON/W0vLK6tp6YaO4ubW9s2vv7Td1nCrKGjQWsWoHRDPBI9YADoK1E8WIDARrBcOrcd66Z0rzOLqDLGGeJP2Ih5wSMJZvl4RP8SV2z3AX2APkWMqRb5edijMRXgR3BmU0U923v7q9mKaSRUAF0brjOgl4OVHAqWCjYjfVLCF0SPqsYzAikmkvnyw/wkfG6eEwVuZEgCfu7xs5kVpnMjCTksBAz2dj87+sk0J44eU8SlJgEZ0+FKYCQ4zHTeAeV4yCyAwQqrjZFdMBUYSC6atoSnDnv7wIzWrFPalUb0/LteunaR0FdIAO0TFy0TmqoRtURw1EUYae0St6sx6tF+vd+piOLlmzCkvoj6zPHx0flDU=</latexit>

lc= 30 mm

<latexit sha1_base64="wjOoKNzkUSfo0fOR9hyq/MsKmWs=">AAAB/HicbZBLSwMxFIUzPmt9jXbpJlgEV2WmFXQjFARxWcE+oB2GTJppQ5OZIbkjDkPFf+LGhSJu/SHu/Demj4W2Hgh8nHNDbk6QCK7Bcb6tldW19Y3NwlZxe2d3b98+OGzpOFWUNWksYtUJiGaCR6wJHATrJIoRGQjWDkZXk7x9z5TmcXQHWcI8SQYRDzklYCzfLgmf4ktcc3AP2APkWMqxb5edijMVXgZ3DmU0V8O3v3r9mKaSRUAF0brrOgl4OVHAqWDjYi/VLCF0RAasazAikmkvny4/xifG6eMwVuZEgKfu7xs5kVpnMjCTksBQL2YT87+sm0J44eU8SlJgEZ09FKYCQ4wnTeA+V4yCyAwQqrjZFdMhUYSC6atoSnAXv7wMrWrFrVWqt2fl+vXTrI4COkLH6BS56BzV0Q1qoCaiKEPP6BW9WY/Wi/VufcxGV6x5hSX0R9bnDxh0lDI=</latexit>

4700 mm

<latexit sha1_base64="r7h0GPUde7OMsKBVB3PF1HAjVXc=">AAAB+HicbZBLSwMxFIUzPmt9dNSlm2ARXJWZWqjLgiAuK9gHtEPJpJk2NMkMyR2xDgX/hxsXirj1p7jz35g+Ftp6IPBxzg25OWEiuAHP+3bW1jc2t7ZzO/ndvf2Dgnt41DRxqilr0FjEuh0SwwRXrAEcBGsnmhEZCtYKR1fTvHXPtOGxuoNxwgJJBopHnBKwVs8tVKqeh7vAHiDDUk56btEreTPhVfAXUEQL1XvuV7cf01QyBVQQYzq+l0CQEQ2cCjbJd1PDEkJHZMA6FhWRzATZbPEJPrNOH0extkcBnrm/b2REGjOWoZ2UBIZmOZua/2WdFKLLIOMqSYEpOn8oSgWGGE9bwH2uGQUxtkCo5nZXTIdEEwq2q7wtwV/+8io0yyX/olS+rRRr10/zOnLoBJ2ic+SjKqqhG1RHDURRip7RK3pzHp0X5935mI+uOYsKj9EfOZ8/hMWSxw==</latexit>

Fig. 2: Schematic of geometry and boundary conditions for a trapezoidal domain and the observed crack

patterns at ¯

u=0.325 mm with different regularization lengths.

0 0.1 0.2 0.3 0.4 0.5

200

400

600

(a) Shape parameter p=2.5.

0 0.1 0.2 0.3 0.4 0.5

200

400

600

(b) Shape parameter p=10.

Fig. 3: The force-displacement curves obtained from the trapezoid problem with different regularization

lengths: non-polar case.

We then repeat the same problem with the micropolar material, i.e., the case where D={B,C,R},397

U=∅. Recall Section 3.5 that the regularization length insensitive response is expected for the micropolar398

material as well. For this problem, we set coupling number N=0.5, and bending characteristic length399

lb=50 mm, while we choose the shape parameter as p=10 which produced regularization length400

insensitive results in Fig. 3(b). As illustrated in Fig. 4, the force-displacement curve conﬁrms that our choice401

p=10 yields the regularization length insensitive global response for micropolar material as well. From402

16 Hyoung Suk Suh et al.

0 0.1 0.2 0.3 0.4 0.5

200

400

600

Fig. 4: The force-displacement curve obtained from the trapezoid problem with different regularization

lengths: micropolar case, p=10, lb=50 mm, and N=0.5.

this numerical example, we again spotlight the fact that the high value of the shape parameter pyields the403

regularization length insensitive response in both non-polar and micropolar material.404

5.2 Single edge notched tests405

We now consider the classical boundary value problem, which serves as a platform to investigate the size406

effect of elasticity and energy dissipation on the crack nucleation and propagation. The problem domain407

is a square plate that has an initial horizontal edge crack placed at the middle from the left to the center408

(Fig. 5). Similar to the previous studies [Miehe et al.,2010a,Borden et al.,2012], we choose material param-409

eters as: E=210 GPa, ν=0.3, Gc=2.7 N/mm, lc=0.008 mm, and ψcrit =10 MJ/m3. Numerical experi-410

ments are conducted with different bending characteristic lengths: lb=0.0, 0.01, 0.05, and 0.25 mm while411

the coupling number is held ﬁxed as N=0.5. Also, for this problem we damage all the energy density412

parts, i.e., D={B,C,R},U=∅. As illustrated in Fig. 5, two different types of simulations are conducted413

with the same specimen: the pure tensile test with prescribed vertical displacement ∆¯

u2=2.0 ×10−5mm;414

and the pure shear test with prescribed horizontal displacement ∆¯

u1=2.0 ×10−5mm. In both cases, the415

displacements are prescribed along the entire top boundary, while the bottom part of the domain is ﬁxed.416

Fig. 6illustrates the crack trajectories at the completely damaged stages for both tension and shear417

tests with different the bending characteristic lengths. The results clearly show that pure tensile loading418

exhibits the same crack pattern regardless of lb. This result is expected, as introducing the micro-polar effect419

should not break the symmetry of the boundary value problems in pure Mode I loading. Interestingly, the420

higher-order constitutive responses have a profound impact on the crack propagation direction in the421

Mode II simulations. As shown in Fig. 6, the micropolar effect leads to a propagation direction bends422

counterclockwise. As the driving force for the phase ﬁeld is affected by the micro-rotation induced by the423

coupling between shear and micro-rotation, this affect the energy dissipation mechanism and ultimately424

the energy minimizer of the action functional that provides the deformed conﬁguration and crack patterns.425

As pointed out in Yavari et al. [2002], the particles near crack tip resist micro-rotation and separation426

(i.e., interlocking), and the crack propagation mechanism in micropolar continuum therefore consists of the427

following steps. First, micro-rotational bonding between adjacent particles at the crack tip breaks and the428

particles starts to rotate with respect to each other. Second, the particles then move apart and the adjacent429

set of particles become the next crack-tip particles. Based on the mechanism, the crack path that minimizes430

the effort on breaking the micro-rotational bond (i.e., the path that maximizes the dissipation) is equivalent431

to the shortest path towards the boundary if the material is isotropic and homogeneous (e.g., a horizontal432

crack growth from the notch tip for the pure tensile loading case). The observed crack patterns shown433

in Fig. 6is consistent with this interpretation. Furthermore, Fig. 6also provides evidence to support that434

Micropolar phase ﬁeld fracture 17

0.5 mm

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0.5 mm

Shear

<latexit sha1_base64="cT340I4kfzNyC4hsLmvcK1AQxzc=">AAAB83icbZBLSwMxFIUz9VXrq+rSTbAIrspMFXRZEMRlRfuAzlAy6W0bmskMyR2xDAV/hRsXirj1z7jz35g+Ftp6IPBxzg25OWEihUHX/XZyK6tr6xv5zcLW9s7uXnH/oGHiVHOo81jGuhUyA1IoqKNACa1EA4tCCc1weDXJmw+gjYjVPY4SCCLWV6InOENr+T7CI2Z3A2B63CmW3LI7FV0Gbw4lMletU/zyuzFPI1DIJTOm7bkJBhnTKLiEccFPDSSMD1kf2hYVi8AE2XTnMT2xTpf2Ym2PQjp1f9/IWGTMKArtZMRwYBaziflf1k6xdxlkQiUpguKzh3qppBjTSQG0KzRwlCMLjGthd6V8wDTjaGsq2BK8xS8vQ6NS9s7KldvzUvX6aVZHnhyRY3JKPHJBquSG1EidcJKQZ/JK3pzUeXHenY/ZaM6ZV3hI/sj5/AGxRZKG</latexit>

Tension

<latexit sha1_base64="/cnOmz5ghbo9yQjUTHCIRl6mm+I=">AAAB9XicbVDLSsNAFJ3UV62vqks3wSK4KkkVdFkQxGWFvqCNZTK9aYdOJmHmRi2h4Ge4caGIW//FnX/jpO1CWw9cOJxzLnPn+LHgGh3n28qtrK6tb+Q3C1vbO7t7xf2Dpo4SxaDBIhGptk81CC6hgRwFtGMFNPQFtPzRVea37kFpHsk6jmPwQjqQPOCMopHuugiPmNZBZoFJr1hyys4U9jJx56RE5qj1il/dfsSSECQyQbXuuE6MXkoVciZgUugmGmLKRnQAHUMlDUF76fTqiX1ilL4dRMqMRHuq/t5Iaaj1OPRNMqQ41IteJv7ndRIMLr2UyzhBkGz2UJAIGyM7q8DucwUMxdgQyhQ3t9psSBVlaIoqmBLcxS8vk2al7J6VK7fnper106yOPDkix+SUuOSCVMkNqZEGYUSRZ/JK3qwH68V6tz5m0Zw1r/CQ/IH1+QNtRpOH</latexit>

Fig. 5: Schematic of geometry and boundary conditions for the single edge notched tests.

the non-polar model is a special case of the micropolar model in which lb≈0. With a sufﬁciently small435

bending characteristic length lb, the difference in crack patterns for the non-polar and micropolar cases436

are negligible. The coupling and the micro-rotational parts (ψC

eand ψR

e) then become signiﬁcant enough to437

play a bigger role for crack growth as the bending characteristic length lbincreases.438

non-polar

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lb=0.0 mm

<latexit sha1_base64="MwpnYLMYuuEWEncQ/RPiV2YKafY=">AAAB/XicbZBLSwMxFIUzPmt9jY+dm2ARXJWZKuhGKLhxWcU+oB2GTJq2oUlmSO6IdSj+FTcuFHHr1t/gzn9j+lho64HA4ZwbcvNFieAGPO/bWVhcWl5Zza3l1zc2t7bdnd2aiVNNWZXGItaNiBgmuGJV4CBYI9GMyEiwetS/HPX1O6YNj9UtDBIWSNJVvMMpARuF7r4II3yBvaKHW8DuIcNSDkO3YIOx8Lzxp6aApqqE7lerHdNUMgVUEGOavpdAkBENnAo2zLdSwxJC+6TLmtYqIpkJsvH2Q3xkkzbuxNoeBXic/r6REWnMQEZ2UhLomdluFP7XNVPonAcZV0kKTNHJQ51UYIjxCAVuc80oiIE1hGpud8W0RzShYIHlLQR/9svzplYq+ifF0vVpoXzzOcGRQwfoEB0jH52hMrpCFVRFFD2gJ/SCXp1H59l5c94nowvOFOEe+iPn4weXrZSy</latexit>

lb=0.01 mm

<latexit sha1_base64="u/h8N1LIBodIkeyWR0y+cHwTY60=">AAAB/nicbZBLSwMxFIUz9VXra1RcuQkWwVWZqYJuhIIbl1XsA9phyKRpG5pkhuSOWIaCf8WNC0XcuvQ3uPPfmD4W2nogcDjnhtx8USK4Ac/7dnJLyyura/n1wsbm1vaOu7tXN3GqKavRWMS6GRHDBFesBhwEayaaERkJ1ogGV+O+cc+04bG6g2HCAkl6inc5JWCj0D0QYYQvsVfyfNwG9gAZlnIUukWbTIQXjT8zRTRTNXS/2p2YppIpoIIY0/K9BIKMaOBUsFGhnRqWEDogPdayVhHJTJBN1h/hY5t0cDfW9ijAk/T3jYxIY4YyspOSQN/Md+Pwv66VQvciyLhKUmCKTh/qpgJDjMcscIdrRkEMrSFUc7srpn2iCQVLrGAh+PNfXjT1csk/LZVvzoqV288pjjw6REfoBPnoHFXQNaqiGqIoQ0/oBb06j86z8+a8T0dzzgzhPvoj5+MHC3yU7Q==</latexit>

lb=0.25 mm

<latexit sha1_base64="+IvcASxsLKpT+jdWPQxD63S7fXo=">AAAB/nicbZBLSwMxFIUzPmt9jYorN8EiuBpmqqIboeDGZRX7gHYYMmmmDU1mhuSOWIaCf8WNC0XcuvQ3uPPfmD4W2nog8HHODbk5YSq4Btf9thYWl5ZXVgtrxfWNza1te2e3rpNMUVajiUhUMySaCR6zGnAQrJkqRmQoWCPsX43yxj1TmifxHQxS5kvSjXnEKQFjBfa+CEJ8iV2nfIbbwB4gx1IOA7vkOu5YeB68KZTQVNXA/mp3EppJFgMVROuW56bg50QBp4INi+1Ms5TQPumylsGYSKb9fLz+EB8Zp4OjRJkTAx67v2/kRGo9kKGZlAR6ejYbmf9lrQyiCz/ncZoBi+nkoSgTGBI86gJ3uGIUxMAAoYqbXTHtEUUomMaKpgRv9svzUC873olTvjktVW4/J3UU0AE6RMfIQ+eogq5RFdUQRTl6Qi/o1Xq0nq03630yumBNK9xDf2R9/AAU2JTz</latexit>

lb=0.05 mm

<latexit sha1_base64="eytw1d0OwY7CIUxVXF1XNkLuSPA=">AAAB/nicbZBLSwMxFIUzPmt9jYorN8EiuCozVdGNUHDjsop9QDsMmTRtQ5PMkNwRy1Dwr7hxoYhbl/4Gd/4b08dCWw8EDufckJsvSgQ34HnfzsLi0vLKam4tv76xubXt7uzWTJxqyqo0FrFuRMQwwRWrAgfBGolmREaC1aP+1aiv3zNteKzuYJCwQJKu4h1OCdgodPdFGOFL7BW9M9wC9gAZlnIYugWbjIXnjT81BTRVJXS/Wu2YppIpoIIY0/S9BIKMaOBUsGG+lRqWENonXda0VhHJTJCN1x/iI5u0cSfW9ijA4/T3jYxIYwYyspOSQM/MdqPwv66ZQuciyLhKUmCKTh7qpAJDjEcscJtrRkEMrCFUc7srpj2iCQVLLG8h+LNfnje1UtE/KZZuTgvl288Jjhw6QIfoGPnoHJXRNaqgKqIoQ0/oBb06j86z8+a8T0YXnCnCPfRHzscPEbiU8Q==</latexit>

micropolar (lb>0.0)

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Shear

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Tension

<latexit sha1_base64="cr9/Sy07ZGlvpCdDDJMhXN/Jryo=">AAAB9XicbVBNT8JAFNziF+IX6tFLIzHxRFo00SOJF4+YUCCBSrbLK2zYbpvdV5U0/A8vHjTGq//Fm//GLXBQcJKXTGbmZd9OkAiu0XG+rcLa+sbmVnG7tLO7t39QPjxq6ThVDDwWi1h1AqpBcAkechTQSRTQKBDQDsY3ud9+AKV5LJs4ScCP6FDykDOKRrrvITxh1gSZB6b9csWpOjPYq8RdkApZoNEvf/UGMUsjkMgE1brrOgn6GVXImYBpqZdqSCgb0yF0DZU0Au1ns6un9plRBnYYKzMS7Zn6eyOjkdaTKDDJiOJIL3u5+J/XTTG89jMukxRBsvlDYSpsjO28AnvAFTAUE0MoU9zcarMRVZShKapkSnCXv7xKWrWqe1Gt3V1W6l59XkeRnJBTck5cckXq5JY0iEcYUeSZvJI369F6sd6tj3m0YC0qPCZ/YH3+AF7Jk1Y=</latexit>

Fig. 6: Fracture patterns for single edge notched tests under different micropolar characteristic length lb.

As illustrated in Fig. 7, separated particles tend to rotate in opposite directions since they are no longer439

interlocked after crack formation. By revisiting Eq. (7) and Eq. (13), notice that the material constant γthat440

relates the ﬁctitious undamaged couple stress ¯mRto the micro-curvature ¯κis proportional to the square of441

the characteristic lengths. The relationship implies that larger characteristic length leads to higher rigidity442

of the micropolar material, so that the separated particles tend to experience greater micro-rotation with443

smaller bending characteristic length.444

18 Hyoung Suk Suh et al.

¯u1=0.0072 mm

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Notch

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lb=0.01 mm

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lb=0.05 mm

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lb=0.25 mm

<latexit sha1_base64="hPk7pTCWvZMHJdWPNawW+toCCH0=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPIWkKnoRCl68WcF+QBvCZrttl+4mYXcillDwr3jxoIhXf4c3/43bNgdtfTDweG+GmXlhIrgG1/22CkvLK6trxfXSxubW9o69u9fQcaooq9NYxKoVEs0Ej1gdOAjWShQjMhSsGQ6vJ37zgSnN4+geRgnzJelHvMcpASMF9oEIQnyFXadyjjvAHiHDUo4Du+w67hR4kXg5KaMctcD+6nRjmkoWARVE67bnJuBnRAGngo1LnVSzhNAh6bO2oRGRTPvZ9PwxPjZKF/diZSoCPFV/T2REaj2SoemUBAZ63puI/3ntFHqXfsajJAUW0dmiXiowxHiSBe5yxSiIkSGEKm5uxXRAFKFgEiuZELz5lxdJo+J4p07l7qxcvc3jKKJDdIROkIcuUBXdoBqqI4oy9Ixe0Zv1ZL1Y79bHrLVg5TP76A+szx/a4pQw</latexit>

Fig. 7: Resultant micro-rotation ﬁeld for single edge notched shear tests where ¯

u1=0.0072 mm.

Fig. 8shows the load-deﬂection curves obtained from both tension and shear tests. The colored curves445

indicate the results with nonzero lb, whereas the transparent gray curves denote the non-polar case. Since446

the force stress can be decomposed into two parts, e.g., ¯

σ=¯

σB+¯

σC, micropolar material that possesses447

a large characteristic length tend to exhibit stiffer response compared to those with smaller characteristic448

lengths, due to the micro-continuum coupling effect. Unlike the tension test results in Fig. 8(a), the reaction449

forces reach their peak values under different strain level from the shearing tests [Fig. 8(b)]. This again450

highlights that the micropolar bending characteristic length affects the crack pattern, which in turn reﬂects451

different global response for the same boundary value problem.452

012345

10-3

0.2

0.4

0.6

(a) Global response from the tension test.

0 0.002 0.004 0.006 0.008 0.01

0.1

0.2

0.3

(b) Global response from the shear test.

Fig. 8: The force-displacement curves from the single edge notched tests with regularization length lc=

0.008 mm.

5.3 Asymmetric notched three-point bending tests453

This section examines a problem originally designed by Ingraffea and Grigoriu [1990], which involves a454

three-point bending of a specimen with three holes. The domain of the problem is illustrated in Fig. 9.455

Since previous studies [Ingraffea and Grigoriu,1990,Miehe et al.,2010a,Patil et al.,2018,Qinami et al.,456

Micropolar phase ﬁeld fracture 19

2019] have shown that different crack patterns can be observed depending on the notch depth and its457

position, we only focus on the case where the notch depth is set to be 25.4 mm.458

32 mm

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51 mm

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51 mm

13 mm

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102 mm

<latexit sha1_base64="qaOG1Fzjc0T6v2o8G2jY9XvSG3c=">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</latexit>

152 mm

<latexit sha1_base64="XC4zfaXEE9DVrSCrYoPACZ4cxvk=">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</latexit>

25.4mm

<latexit sha1_base64="53xbneY4DyH2SGcMlxAoaD1vH/Y=">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</latexit>

457 mm

<latexit sha1_base64="HBsZAAf+b4LSnUnARRB0Qbtw8sA=">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</latexit>

508 mm

<latexit sha1_base64="6uwxzsKOMBPoSIbNbh1btsJdQqg=">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</latexit>

203 mm

<latexit sha1_base64="8ZSaCUcRGdf6HoII7mY5iMyzrs8=">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</latexit>

<latexit sha1_base64="ZZCWsiRu0pgt5O9epAZtvNr/dkU=">AAAEdnicjVNbaxNBFJ42UWu8pfooSDSIPpWsBPSxYKG+qFVMG+jEMDt7Nh0zl2VmNrUMA/4QX/U/+U98dCabQrpxxYE9HL7znW9mzyUtODN2MPi1td1qX7t+Y+dm59btO3fvdXfvHxtVagojqrjS45QY4EzCyDLLYVxoICLlcJLOX8f4yQK0YUp+shcFTASZSZYzSmyApt0uTol22MJXm+au9H7a7Q/2BsvT23SSldNHq3M03W29w5mipQBpKSfGOKItoxx8r4NLAwWhczKD09LmryaOyaK0IKlfjzkijCD2bAPMlbRmAzUXIr0KxkIxOatRo6RViv8F1iavoZYJqEE5V8QGSMI5VUIQmcVCSaO0P00mDge31BDlHE4Vz+K7FHf9xPta1pdSFFUOT3VQx49XTq+f9LC+xCqnlks4aFslxy4tW+40ZH55UQcfQKi6hrfhGe8L0MSGMJ5pEghYkpSTJk547jrtc6XvjG9UzVgYpUs+ppmyTczDunCsWm/cqHywrrye8M9bwh94F01TfH4e4sE0xG24MnxNfwsL76JpUp/JoB7Msl/2DFQYBqfF3LuPYSp0bKSG9VYugP7H5IQNTOr7tukcv9hLhnvDD8P+/uG3ahd30EP0BD1HCXqJ9tEbdIRGiKIF+o5+oJ+t3+1H7aftZxV1e2u1vw/QldMe/AHKf5LS</latexit>

102 mm

Fig. 9: Schematic of geometry and boundary conditions for the three-point bending tests.

We consider a specimen composed of a micropolar material and choose Boltzmann material parameters459

close to the properties of Plexiglas specimen tested by Ingraffea and Grigoriu [1990]: E=3.2 GPa, ν=0.3,460

Gc=0.31 N/mm, lc=1.0 mm, and ψcrit =7.5 kJ/m3. For this problem, we assume that all the energy461

density parts can be degraded, i.e., D={B,C,R},U=∅.462

Within the problem domain (Fig. 9), we ﬁrst attempt to investigate the effect of coupling number on the463

crack trajectory by conducting multiple numerical tests with different values: N=0.1, N=0.5, and N=464

0.9, while bending characteristic length is held ﬁxed as lb=10.0 mm. The numerical simulation conducted465

under a displacement-controlled regime where we keep the load increment as ∆¯

u2=−2.0 ×10−4mm.466

Fig. 10 illustrates the crack trajectories obtained by numerical experiments with different coupling num-467

ber Nin comparison to the experimental result [Ingraffea and Grigoriu,1990], while Fig. 11 shows the mea-468

sured reaction forces as a function of crack-mouth-opening-displacement (CMOD). Similar to the experi-469

mental results in Fig. 10(a), numerical results [Fig. 10(b)-(d)] show that the cracks tend to deﬂect towards470

the holes, eventually coalescing with the intermediate one. However, taking a closer look at Fig. 10(b)-(d),471

one can observe the slight differences on the concavity of the crack topologies especially when the crack472

passes close to the bottom hole. As we highlighted in Section 5.2, the crack tends to propagate through the473

path where the energy that takes to break the micro-rotational bond is minimized. At the same time, based474

on Saint-Venant’s principle, the crack trajectory can also be affected by the bottom hole due to an increase475

of the singularity [Qinami et al.,2019]. In this speciﬁc platform, it can thus be interpreted that the crack476

propagation may result from the competition between the two, based on the obtained results. Since the477

material that possesses higher degree of micropolarity requires more energy to break the micro-rotational478

bond, the bottom hole effect on the crack trajectory becomes negligible as coupling number Nincreases479

[Fig. 10(b)-(d)]. In addition, Fig. 11 implies that if more energy is required to break the micro-rotational480

bond, it results in higher material stiffness in the elastic regime, supporting our interpretation.481

We then conduct a brief sensitivity analysis with respect to the time discretization (i.e., prescribed dis-482

placement increment ∆¯

u2) within the same problem domain, while we set the coupling number to be483

N=0.9 during the analysis. Fig. 12 illustrates the simulated crack patterns for asymmetrically notched484

beam with three holes, with different prescribed displacement rates, varying from ∆¯

u2=−4.0 ×10−4mm485

to −16.0 ×10−4mm. Since meaningful differences in the crack trajectories are not observed, the result con-486

ﬁrms the practical applicability of the explicit operator-splitting solution scheme, if the load increment is487

small enough.488

20 Hyoung Suk Suh et al.

(b) N=0.1

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<latexit sha1_base64="5olRcktIuZGF/llud13lrU9TIcc=">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</latexit>

(d) N=0.9

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(a)

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Fig. 10: Crack topologies for asymmetric notched three-point bending test: (a) experimentally obtained

pattern by Ingraffea and Grigoriu [1990]; (b)-(d) numerically obtained parttern with different coupling

number N.

0123

0.1

0.2

0.3

0.4

Fig. 11: Load-CMOD curves from the three-point bending test.

5.4 Double edge notched tests489

This numerical example investigates the effect of partial degradation of the strain energy density on the490

crack patterns. As illustrated in Fig. 13, the problem domain is a 100 mm wide and 100 mm long square491

plate with two 25 mm long symmetric initial horizontal edge notches at the middle. We assign the following492

material properties for this problem: E=30 GPa, ν=0.2, Gc=0.1 N/mm, lc=0.75 mm, and ψcrit =1.0493

kJ/m3. Numerical experiments are simulated with bending characteristic length lb=30.0 mm and the494

coupling number is set to be N=0.5. While the bottom part of the domain is held ﬁxed, we prescribe the495

displacement along the entire top boundary at an angle of 45 degrees to the horizontal direction: ∆¯

u1=496

∆¯

u2=5.0 ×10−4mm, such that the domain is subjected to combined tensile and shear loads.497

Regarding partial degradation, recall Eq. (25) that energy density parts that corresponds to the set U498

remains completely undamaged, such that gi(d) = 1 for i∈U, where D∪U={B,C,R}and D∩U=∅.499

Within this platform, we ﬁrst explore the effects of each individual energy density part by considering500

Micropolar phase ﬁeld fracture 21

(a)

(b)

<latexit sha1_base64="oF5R3DDGiloRLmNVK9ayt+oCgC4=">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</latexit>

(c)

<latexit sha1_base64="qlq0tFj8kMnuWQp51Gt4x+cItWc=">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</latexit>

(d)

<latexit sha1_base64="s+ILh8goAt2n9H6wE19HIbklKyk=">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</latexit>

Fig. 12: Observed crack patterns for asymmetric notched three-point bending tests: (a) ∆¯

u2=−2.0 ×10−4

mm; (b) ∆¯

u2=−4.0 ×10−4mm; (c) ∆¯

u2=−8.0 ×10−4mm; (d) ∆¯

u2=−16.0 ×10−4mm.

100 mm

<latexit sha1_base64="4odvApqtH7vRUBGGUVkVT+RyZbA=">AAAB9XicbZBLSwMxFIXv+Kz1VXXpJlgEV2WmCrosCOKygn1AO5ZMmmlDk5khuaOWoeDPcONCEbf+F3f+G9PHQlsPBD7OuSE3J0ikMOi6387S8srq2npuI7+5tb2zW9jbr5s41YzXWCxj3Qyo4VJEvIYCJW8mmlMVSN4IBpfjvHHPtRFxdIvDhPuK9iIRCkbRWnee65I28kfMiFKjTqHoltyJyCJ4MyjCTNVO4avdjVmqeIRMUmNanpugn1GNgkk+yrdTwxPKBrTHWxYjqrjxs8nWI3JsnS4JY21PhGTi/r6RUWXMUAV2UlHsm/lsbP6XtVIML/xMREmKPGLTh8JUEozJuALSFZozlEMLlGlhdyWsTzVlaIvK2xK8+S8vQr1c8k5L5ZuzYuXqaVpHDg7hCE7Ag3OowDVUoQYMNDzDK7w5D86L8+58TEeXnFmFB/BHzucPjPWSUg==</latexit>

25 mm

<latexit sha1_base64="gIdtPmS3mbop8kEGbWy7COgjf+M=">AAAB9HicbZBLSwMxFIXv1Fetr6pLN8EiuCozVdFlQRCXFewD2qFk0rQNTWbG5E6xDAX/hRsXirj1x7jz35g+Ftp6IPBxzg25OUEshUHX/XYyK6tr6xvZzdzW9s7uXn7/oGaiRDNeZZGMdCOghksR8ioKlLwRa05VIHk9GFxP8vqQayOi8B5HMfcV7YWiKxhFa/mlC9JC/ogpUWrczhfcojsVWQZvDgWYq9LOf7U6EUsUD5FJakzTc2P0U6pRMMnHuVZieEzZgPZ402JIFTd+Ol16TE6s0yHdSNsTIpm6v2+kVBkzUoGdVBT7ZjGbmP9lzQS7V34qwjhBHrLZQ91EEozIpAHSEZozlCMLlGlhdyWsTzVlaHvK2RK8xS8vQ61U9M6KpbvzQvnmaVZHFo7gGE7Bg0sowy1UoAoMHuAZXuHNGTovzrvzMRvNOPMKD+GPnM8fJiOSHg==</latexit>

45

50 mm

<latexit sha1_base64="WKAfSdtSmxeLPDsXutJZZV6FXLI=">AAAB9HicbZBLSwMxFIXv1Fetr6pLN8EiuCozVdFlQRCXFewD2qFk0rQNTWbG5E6xDAX/hRsXirj1x7jz35g+Ftp6IPBxzg25OUEshUHX/XYyK6tr6xvZzdzW9s7uXn7/oGaiRDNeZZGMdCOghksR8ioKlLwRa05VIHk9GFxP8vqQayOi8B5HMfcV7YWiKxhFa/kXLmkhf8SUKDVu5wtu0Z2KLIM3hwLMVWnnv1qdiCWKh8gkNabpuTH6KdUomOTjXCsxPKZsQHu8aTGkihs/nS49JifW6ZBupO0JkUzd3zdSqowZqcBOKop9s5hNzP+yZoLdKz8VYZwgD9nsoW4iCUZk0gDpCM0ZypEFyrSwuxLWp5oytD3lbAne4peXoVYqemfF0t15oXzzNKsjC0dwDKfgwSWU4RYqUAUGD/AMr/DmDJ0X5935mI1mnHmFh/BHzucPIwiSHA==</latexit>

50 mm

Fig. 13: Schematic of geometry and boundary conditions for the double edge notched tests.

three different settings: (a) D={B}; (b) D={C}; and (c) D={R}. Fig. 14 shows the fracture patterns501

for double edge notched tests for three aforementioned cases where ¯

u1=¯

u2=0.05 mm. Under the502

same threshold energy density ψcrit, partial Boltzmann degradation case shown in Fig. 14(a) undergoes503

crack propagation, whereas degrading ψC

e[Fig. 14(b)] or ψR

e[Fig. 14(c)] exhibit small amount of damage504

accumulation at the crack tip, without complete rupturing. It reveals that in either force or displacement-505

driven setup, most of the elastic energy is stored in non-polar constituent; the amount of stored energy506

density parts: ψB

e>ψC

e>ψR

e. As illustrated in Fig. 15, only the ﬁctitious undegraded Boltzmann energy507

density part locally exceeds the prescribed threshold ψcrit =1.0 kJ/m3through the cracks when D={B}.508

Even though ψC

eand ψR

edo not exceed the threshold energy except the ﬂaw tip region, the results also509

indirectly evidence that the coupling and pure micro-rotational energy density parts affect crack kinking,510

while the pure Boltzmann part mainly drives the crack to grow.511

Since the pure Boltzmann part mainly drives the crack propagation, we now focus on the combined512

partial degradation with B∈D, also by considering three different settings within the same platform: (a)513

22 Hyoung Suk Suh et al.

D={B}

<latexit sha1_base64="i0C/iJy33SKy3AjYuBxRPF7wTTA=">AAACFHicbVDLSgMxFM3UV62vqks3wSIIQpmpgm6EoiIuK9gHdErJpHfa0MyD5I5QhoK/4MZfceNCEbcu3Pk3po+Fth4InJxzL8k5XiyFRtv+tjILi0vLK9nV3Nr6xuZWfnunpqNEcajySEaq4TENUoRQRYESGrECFngS6l7/cuTX70FpEYV3OIihFbBuKHzBGRqpnT9yA4Y9X7F+ejWk59SV4KMrPcU40AvqKtHtoavG93a+YBftMeg8caakQKaotPNfbifiSQAhcsm0bjp2jK2UKRRcwjDnJhpixvusC01DQxaAbqXjUEN6YJQO9SNlToh0rP7eSFmg9SDwzOQogp71RuJ/XjNB/6yVijBOEEI+echPJMWIjhqiHaGAoxwYwrgS5q+U95gpAE2POVOCMxt5ntRKRee4WLo9KZSvHyZ1ZMke2SeHxCGnpExuSIVUCSeP5Jm8kjfryXqx3q2PyWjGmla4S/7A+vwBaX2e1w==</latexit>

(a)

D={C}

<latexit sha1_base64="Lo2d0A/lnJ5K/drliSIHO1Jm30Q=">AAACFHicbVDLSgMxFM3UV62vqks3wSIIQpmpgm6EQkVcVrAP6JSSSe+0oZkHyR2hDAV/wY2/4saFIm5duPNvTB8LbT0QODnnXpJzvFgKjbb9bWWWlldW17LruY3Nre2d/O5eXUeJ4lDjkYxU02MapAihhgIlNGMFLPAkNLxBZew37kFpEYV3OIyhHbBeKHzBGRqpkz9xA4Z9X7FBejWil9SV4KMrPcU40Ap1lej10VWTeydfsIv2BHSRODNSIDNUO/kvtxvxJIAQuWRatxw7xnbKFAouYZRzEw0x4wPWg5ahIQtAt9NJqBE9MkqX+pEyJ0Q6UX9vpCzQehh4ZnIcQc97Y/E/r5Wgf9FORRgnCCGfPuQnkmJExw3RrlDAUQ4NYVwJ81fK+8wUgKbHnCnBmY+8SOqlonNaLN2eFcrXD9M6suSAHJJj4pBzUiY3pEpqhJNH8kxeyZv1ZL1Y79bHdDRjzSrcJ39gff4Aaw+e2A==</latexit>

(b)

D={R}

<latexit sha1_base64="kImzKFzgXLH8ZKiqp/318UfA3U8=">AAACFHicbVDLSgMxFM3UV62vqks3wSIIQpmpgm6EgiIuq9gHdErJpHfa0MyD5I5QhoK/4MZfceNCEbcu3Pk3po+Fth4InJxzL8k5XiyFRtv+tjILi0vLK9nV3Nr6xuZWfnunpqNEcajySEaq4TENUoRQRYESGrECFngS6l7/YuTX70FpEYV3OIihFbBuKHzBGRqpnT9yA4Y9X7F+ejmk59SV4KMrPcU40FvqKtHtoavG93a+YBftMeg8caakQKaotPNfbifiSQAhcsm0bjp2jK2UKRRcwjDnJhpixvusC01DQxaAbqXjUEN6YJQO9SNlToh0rP7eSFmg9SDwzOQogp71RuJ/XjNB/6yVijBOEEI+echPJMWIjhqiHaGAoxwYwrgS5q+U95gpAE2POVOCMxt5ntRKRee4WLo5KZSvHiZ1ZMke2SeHxCGnpEyuSYVUCSeP5Jm8kjfryXqx3q2PyWjGmla4S/7A+vwBgp2e5w==</latexit>

(c)

Fig. 14: Crack patterns for double edge notched tests obtained by considering different degradation func-

tions [i.e., gi(d) = g(d)if i∈D;gi(d) = 1 otherwise] on each energy density part, where ¯

u1=¯

u2=0.05

mm.

D={B}

(a) ψB

ein kJ/m3

D={C}

(b) ψC

ein kJ/m3

D={R}

ein kJ/m3

Fig. 15: Fictitious undegraded energy density part ψi

e(i∈D), where ¯

u1=¯

u2=0.05 mm.

D={B,R}; (b) D={B,C}; and (c) D={B,C,R}. Fig. 16 shows the crack patterns for double edge514

notched tests for three different combinations of partial degradation compared with the case where D=515

{B}, while Fig. 17 illustrates the obtained load-deﬂection curves. The results conﬁrms that degradation of516

the energy density parts ψC

eand ψR

eaffects the crack kinking and curving.517

The combined degradation with D={B,R}tend to stimulate similar fracture patterns compared to518

the partial Boltzmann degradation case until ¯

u1=¯

u2=0.04 mm, and then the cracks start to propagate519

towards the notches. Revisiting Fig. 15, this again indicates that the crack trajectories tend to follow the520

path that maximizes the energy dissipation (i.e., crack growth towards the adjacent ﬂaw when the stored521

ψR

eat the tip becomes high enough). Similarly, the combined degradation with D={B,C}leads the cracks522

to grow towards the adjacent tip. In this case, however, the cracks tend to kink towards the adjacent notch523

from the beginning, and then two cracks coalescence toward each other after sufﬁcient loading. Since the524

amount of stored coupling energy ψC

eis greater than the pure micro-rotational part ψR

e, we speculate that525

the coupling energy part inﬂuences the crack pattern more signiﬁcantly compared to the micro-rotational526

energy part, so that we thus observe similar fracture pattern when D={B,C,R}. In summary, this nu-527

Micropolar phase ﬁeld fracture 23

¯u1=¯u2

=0.0125 mm

<latexit sha1_base64="Ux6R9oa3Uouaon3gs0luaaT/qt0=">AAACF3icbZBLSwMxFIUzPmt9VV26CRaLqzJTFd0UCm5cVrEP6AxDJk3b0GRmSO6IZei/cONfcaOgiFvd+W9MH4K2Hgh8nHNDck8QC67Btr+shcWl5ZXVzFp2fWNzazu3s1vXUaIoq9FIRKoZEM0ED1kNOAjWjBUjMhCsEfQvRnnjlinNo/AGBjHzJOmGvMMpAWP5uWLBDYhKk6Hv4DL+4RJ23WyhjO2i7ZROsQvsDlIs5dDP5Y03Fp4HZwp5NFXVz3267YgmkoVABdG65dgxeClRwKlgw6ybaBYT2idd1jIYEsm0l473GuJD47RxJ1LmhIDH7u8bKZFaD2RgJiWBnp7NRuZ/WSuBzrmX8jBOgIV08lAnERgiPCoJt7liFMTAAKGKm79i2iOKUDBVZk0JzuzK81AvFZ3jYunqJF+5fp7UkUH76AAdIQedoQq6RFVUQxTdo0f0gl6tB+vJerPeJ6ML1rTCPfRH1sc3sM+dyQ==</latexit>

¯u1=¯u2

=0.025 mm

<latexit sha1_base64="Gg730hbC7blqJ78xcRl1C5Y8nv4=">AAACFnicbZBLSwMxFIUz9VXrq+rSTbBY3FhmRkU3hYIbl1XsAzqlZNK0DU1mhuSOWIb+Cjf+FTciirgVd/4b04egrQcCH+fckNzjR4JrsO0vK7WwuLS8kl7NrK1vbG5lt3eqOowVZRUailDVfaKZ4AGrAAfB6pFiRPqC1fz+xSiv3TKleRjcwCBiTUm6Ae9wSsBYrexR3vOJSuJhy8FF/MMu9rxMvojtgu2eYg/YHSRYymErmzPWWHgenCnk0FTlVvbTa4c0liwAKojWDceOoJkQBZwKNsx4sWYRoX3SZQ2DAZFMN5PxWkN8YJw27oTKnADw2P19IyFS64H0zaQk0NOz2cj8L2vE0DlvJjyIYmABnTzUiQWGEI86wm2uGAUxMECo4uavmPaIIhRMkxlTgjO78jxU3YJzXHCvTnKl6+dJHWm0h/bRIXLQGSqhS1RGFUTRPXpEL+jVerCerDfrfTKasqYV7qI/sj6+ATUAnY4=</latexit>

¯u1=¯u2

=0.05 mm

<latexit sha1_base64="rPlYNB5As5cuS7uPwlh1ObCN2aQ=">AAACFXicbZBLSwMxFIUz9VXra9Slm2CxuJAyUxXdFApuXFaxD+iUkknTNjSZGZI7Yhn6J9z4V9wIKuJWcOe/MX0I2nog8HHODck9fiS4Bsf5slILi0vLK+nVzNr6xuaWvb1T1WGsKKvQUISq7hPNBA9YBTgIVo8UI9IXrOb3L0Z57ZYpzcPgBgYRa0rSDXiHUwLGatlHOc8nKomHLRcX8Q8XsOdlckXs5J1T7AG7gwRLOWzZWeOMhefBnUIWTVVu2Z9eO6SxZAFQQbRuuE4EzYQo4FSwYcaLNYsI7ZMuaxgMiGS6mYy3GuID47RxJ1TmBIDH7u8bCZFaD6RvJiWBnp7NRuZ/WSOGznkz4UEUAwvo5KFOLDCEeFQRbnPFKIiBAUIVN3/FtEcUoWCKzJgS3NmV56FayLvH+cLVSbZ0/TypI4320D46RC46QyV0icqogii6R4/oBb1aD9aT9Wa9T0ZT1rTCXfRH1sc3t8WdUg==</latexit>

D={B}

<latexit sha1_base64="nJk68jbpHPramtoJAM6fR89kydY=">AAACFHicbVDLSgMxFM34dnyNunQTLIIglBkVdCMUdeFSwarQKSWT3mlDMw+SO0IZCv6CG3/FjQtF3Lpw59+YmXahrQcCJ+fcS3JOkEqh0XW/ranpmdm5+YVFe2l5ZXXNWd+40UmmONR5IhN1FzANUsRQR4ES7lIFLAok3Aa9s8K/vQelRRJfYz+FZsQ6sQgFZ2iklrPnRwy7oWK9/Hxgn9i+hBB9GSjGgZ5SX4lOF31V3ltOxa26Jegk8UakQka4bDlffjvhWQQxcsm0bnhuis2cKRRcwsD2Mw0p4z3WgYahMYtAN/My1IDuGKVNw0SZEyMt1d8bOYu07keBmSwi6HGvEP/zGhmGx81cxGmGEPPhQ2EmKSa0aIi2hQKOsm8I40qYv1LeZaYAND3apgRvPPIkudmvegfV/avDSu36YVjHAtki22SXeOSI1MgFuSR1wskjeSav5M16sl6sd+tjODpljSrcJH9gff4AJgeeuQ==</latexit>

D={B,C,R}

<latexit sha1_base64="sxth43kokL9+CDTSGKw5BN68rmw=">AAACGHicbZBLSwMxFIUzPuv4qrp0EyyCC6kzVdCNUKwLl1WsCp1SMumdNjTzILkjlKHgn3DjX3HjQhG33flvTB8Lbb0QOJxzQ3I+P5FCo+N8W3PzC4tLy7kVe3VtfWMzv7V9p+NUcajxWMbqwWcapIighgIlPCQKWOhLuPe7lWF+/whKizi6xV4CjZC1IxEIztBYzfyRFzLsBIp1s8u+fW57EgL0pK8YB3pxWDm8oZ4S7Q56auQ18wWn6IyGzgp3IgpkMtVmfuC1Yp6GECGXTOu66yTYyJhCwSX0bS/VkDDeZW2oGxmxEHQjGxXr033jtGgQK3MipCP3942MhVr3Qt9sDmvo6Wxo/pfVUwzOGpmIkhQh4uOHglRSjOmQEm0JBRxlzwjGlTB/pbzDDAA0LG0DwZ2uPCvuSkX3uFi6PimUb5/GOHJkl+yRA+KSU1ImV6RKaoSTZ/JK3smH9WK9WZ/W13h1zpog3CF/xhr8AFqBn84=</latexit>

D={B,R}

<latexit sha1_base64="tL9A5tsF7T/vleGGZ2AlluQ8GDI=">AAACFnicbZBLSwMxFIUzPuv4qrp0EyyCCy0zVdCNUNSFyyp9CJ1SMumdNjTzILkjlKHgf3DjX3HjQhG34s5/Y/pY+LoQOJxzQ3I+P5FCo+N8WjOzc/MLi7kle3lldW09v7FZ13GqONR4LGN14zMNUkRQQ4ESbhIFLPQlNPz++Shv3ILSIo6qOEigFbJuJALBGRqrnT/wQoa9QLF+djG0T21PQoCe9BXjQM/2r6mnRLeHnho77XzBKTrjoX+FOxUFMp1KO//hdWKehhAhl0zrpusk2MqYQsElDG0v1ZAw3mddaBoZsRB0KxvXGtJd43RoECtzIqRj9/uNjIVaD0LfbI5K6N/ZyPwva6YYnLQyESUpQsQnDwWppBjTESPaEQo4yoERjCth/kp5jxkAaEjaBoL7u/JfUS8V3cNi6eqoUK7eTXDkyDbZIXvEJcekTC5JhdQIJ/fkkTyTF+vBerJerbfJ6ow1RbhFfoz1/gVLmJ9L</latexit>

D={B,C}

<latexit sha1_base64="3no5Ph8QTyl1W30kPuaowg0qSwY=">AAACFnicbZBLSwMxFIUzPuv4qrp0EyyCCy0zKuhGEOvCZYVWhc5QMumdNjTzILkjlKHgf3DjX3HjQhG34s5/YzrtwteFwOGcG5LzBakUGh3n05qanpmdmy8t2ItLyyur5bX1K51kikOTJzJRNwHTIEUMTRQo4SZVwKJAwnXQr43y61tQWiRxAwcp+BHrxiIUnKGx2uU9L2LYCxXr5+dD+8T2JIToyUAxDvRst0Y9Jbo99FThtMsVp+oUQ/8KdyIqZDL1dvnD6yQ8iyBGLpnWLddJ0c+ZQsElDG0v05Ay3mddaBkZswi0nxe1hnTbOB0aJsqcGGnhfr+Rs0jrQRSYzVEJ/Tsbmf9lrQzDYz8XcZohxHz8UJhJigkdMaIdoYCjHBjBuBLmr5T3mAGAhqRtILi/K/8VV/tV96C6f3lYOW3cjXGUyCbZIjvEJUfklFyQOmkSTu7JI3kmL9aD9WS9Wm/j1SlrgnCD/Bjr/Qs0Cp88</latexit>

Fig. 16: Crack patterns for double edge notched tests with different combination of degrading energy den-

sity parts at several load increments.

merical experiment highlight that the pure Boltzmann energy density drives the crack growth, while the528

micro-continuum coupling energy density mainly inﬂuences the kinking direction.529

6 Conclusion530

This study presents a phase ﬁeld fracture framework to model cohesive fracture in micropolar continua.531

To the best of the authors’ knowledge, this is the ﬁrst ever mathematics model that employs the phase532

ﬁeld fracture framework to simulate crack growth in materials that exhibit size effect in both elastic and533

damaged regimes. To replicate a consistent size effect for both the elastic deformation and crack growth534

mechanisms, we introduce a method to incorporate distinctive degradation mechanisms via an energy-535

split approach for the non-polar, coupling and micropolar energies, while adopting the pair of degradation536

and regularization proﬁles that enables us to suppress the sensitivity of the length scale parameter that reg-537

ularizes the phase ﬁeld. One-dimensional analysis and numerical experiments demonstrate that the quasi-538

quadratic degradation function combined with linear local dissipation function successfully suppress the539

sensitivity of the length scale parameter for phase ﬁeld while successfully incorporate the size effect with540

a length scale parameter that can be measured via standard inverse problems for micropolar materials.541

This result is signiﬁcant, as the insensitivity of the length scale parameter will allow one to use coarser542

mesh to run simulations for a scale relevant to ﬁeld applications (e.g. geological formations, structural543

components), while still able to replicating the size effect exhibited by materials of internal structures.544

24 Hyoung Suk Suh et al.

0 0.02 0.04 0.06

0.05

0.1

0.15

D={C}

Fig. 17: The force-displacement curves from the double edge notched tests with different combination of

degrading energy density parts.

7 Acknowledgments545

The authors would like to thank two anonymous reviewers who have provided helpful suggestions and546

feedback that improved the manuscript. The ﬁrst and corresponding authors are supported by the Earth547

Materials and Processes program from the US Army Research Ofﬁce under grant contract W911NF-18-2-548

0306. The corresponding author is supported by the NSF CAREER grant from Mechanics of Materials and549

Structures program at National Science Foundation under grant contract CMMI-1846875, the Dynamic550

Materials and Interactions Program from the Air Force Ofﬁce of Scientiﬁc Research under grant contracts551

FA9550-17-1-0169 and FA9550-19-1-0318. These supports are gratefully acknowledged. The views and con-552

clusions contained in this document are those of the authors, and should not be interpreted as representing553

the ofﬁcial policies, either expressed or implied, of the sponsors, including the Army Research Laboratory554

or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Gov-555

ernment purposes notwithstanding any copyright notation herein.556

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Dec 2024

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Apr 2025

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Feb 2025

In this study, we introduce a novel stretch-based gradient-enhanced damage (GED) model that allows the fracture to localize and also captures the development of a physically diffuse damage zone. This capability contrasts with the paradigm of the phase field method for fracture, where a sharp crack is numerically approximated in a diffuse manner. Capturing fracture localization and diffuse damage in our approach is achieved by considering nonlocal effects that encompass network topology, heterogeneity, and imperfections. These considerations motivate the use of a statistical damage function dependent upon the nonlocal deformation state. From this model, fracture toughness is realized as an output. While GED models have been classically utilized for damage modeling of structural engineering materials, they face challenges when trying to capture the cascade from damage to fracture, often leading to damage zone broadening. This deficiency contributed to the popularity of the phase-field method over the GED model for elastomers and other quasi-brittle materials. Other groups have proceeded with damage-based GED formulations that prove identical to the phase-field method (Lorentz et al., 2012), but these inherit the aforementioned limitations. To address this issue in a thermodynamically consistent framework, we implement two modeling features (a nonlocal driving force bound and a simple relaxation function) specifically designed to capture the evolution of a physically meaningful damage field and the simultaneous localization of fracture, thereby overcoming a longstanding obstacle in the development of these nonlocal strain- or stretch-based approaches. We discuss several numerical examples to understand the features of the approach at the limit of incompressibility, and compare them to the phase-field method as a benchmark for the macroscopic response and fracture energy predictions.

Inference of phase field fracture models

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Full-text available

Apr 2025

The phase field approach to modeling fracture uses a diffuse damage field to represent a crack. This addresses the singularities that arise at the crack tip in computations with sharp interface models, mollifying some of the difficulties associated with the mathematical and numerical treatment of fracture. The introduction of the diffuse field helps with crack propagation dynamics, enabling phase-field approaches to model all phases of damage from crack initiation to propagation, branching, and merging. Specific formulations, beginning with brittle fracture, have also been shown to converge to classical solutions. Extensions to cover the range of material failure, including ductile and cohesive fracture, leads to an array of possible models. There exists a large body of studies of these models and their consequences for crack evolution. However, there have not been systematic studies into how optimal models may be chosen. Here we take a first step in this direction by developing formal methods for identification of the best parsimonious model of phase field fracture given full-field data on the damage and deformation fields. We consider some of the main models that have been used to model damage, its degradation of elastic response, and its propagation. Our approach builds upon Variational System Identification (VSI), a weak form variant of the Sparse Identification of Nonlinear Dynamics (SINDy). In this first communication we focus on synthetically generated data but we also consider central issues associated with the use of experimental full-field data, such as data sparsity and noise.

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In this article, we develop a peridynamic model for the micropolar elastic solids called operator-derived micropolar peridynamics (OMPD). Two types of OMPD models, namely the OMPD model-I based on the peridynamic differential operator and model-II based on the peridynamic operator method with second-order vector derivative, are obtained. By using the first-order Taylor series expansion (TSE), model-I can recover the previously proposed non-ordinary state-based micropolar peridynamics. However, the OMPD Model-I of any order TSE still suffers from zero-energy mode instability, while the proposed OMPD model-II is free of zero-energy mode and thus produces the correct micropolar elasticity response. The OMPD model-II produces an ordinary-like state-based micropolar peridynamic model with a symmetric horizon, which degenerates to the ordinary state-based peridynamics by vanishing the Cosserat shear modulus and micro-rotations. Furthermore, a novel bond-based micropolar peridynamics is derived with some moduli restrictions. Such bond-based micropolar peridynamics considers shear deformability and average micro-rotational effect in the shear bonds, and inherits the couple force generated by shear bond force from the OMPD model-II, which is essential to be consistent with Eringen’s micropolar elastic theory. We show that the novel bond-based micropolar peridynamics relaxes Poisson’s ratio fixation to a rigorous range from -1 to 1/4. Several numerical examples are provided to validate the models’ capacity in modeling micropolar solids and the crack propagation behavior.

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Nov 2024
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Nov 2024
INT J MECH SCI

This article proposes a micropolar phase-field model for size-dependent brittle fracture in solids under electro-mechanical loading conditions. Considering displacement, micro-rotation, electric potential, and phase-field variable as the kinematic descriptors and employing the virtual power principle, we derive a set of coupled governing partial differential equations (PDEs) for size-dependent solids. Invoking the first and second laws of thermodynamics, we determine the constitutive relations for the thermodynamic fluxes. Carrying out the finite element implementation of the derived governing PDEs using the open-source Gridap package in Julia, we demonstrate the efficacy of the proposed phase-field model through a few representative numerical examples. Especially the importance of the proposed model in incorporating the effect of relative rotation, i.e., the difference between macro- and micro-rotation, on the response of solids under electro-mechanical loading is shown that may not be possible with the existing non-local models such as strain-gradient or couple-stress approaches. To capture the experimentally observed size effects in solids under electro-mechanical loading, the proposed model does not demand higher-order continuity of the field variables, unlike a typically used strain gradient model. To demonstrate the efficacy of the proposed model, we have compared our results against demanding experimental and numerical benchmark results available in the literature. We provide a parametric study to unravel the effect of different micropolar material parameters on the electro-mechanical response of a brittle solid. Interestingly, the proposed micropolar model is less sensitive to the phase-field length scale than the conventional non-polar phase-field models.

A microscale model for concrete failure in poro-elasto-plastic media

Article

Full-text available

Feb 2020
THEOR APPL FRACT MEC

Fadi Aldakheel

This work provides a micromechanical framework for modeling water-induced failure mechanisms of concrete in an experimental-virtual lab. The complicated geometry and content of concrete at a lower scale can be detected by a computed tomography (micro-CT) scan. Based on the experimental observations, we developed a constitutive model for the coupled problem of fluid-saturated heterogeneous porous media at fracture. The poro-plasticity model is additively decomposed into reversible-elastic and irreversible-plastic parts. The governing formulations are based on an energetic response function and a dissipated work due to plasticity (Drucker-Prager model), fluid transport (Darcy’s law) and fracture (phase-field method) for the multiphysics problem. The model performance is demonstrated through some representative examples in 2D, representing an idealized microstructure of concrete.

An open-source FEniCS implementation of a phase field fracture model for micropolar continua

Article

Full-text available

Dec 2019
INT J MULTISCALE COM

A micropolar phase field fracture model is implemented in an open-source library FEniCS. This implementation is based on the theoretical study in Suh et al. [2020] in which the resultant phase field model exhibits the consistent micropolar size effect in both elastic and damage regions identifiable via inverse problems for micropolar continua. By leveraging the automatic code generation technique in FEniCS, we provide documentation of the source code expressed in a language very close to the mathematical expressions without comprising significant efficiency. This combination of generality and interpretability, therefore, enables us to provide a detailed walk-through that connects the implementation with the regularized damage theory for micropolar materials. By making the source code open source, the paper will provide an efficient development and educational tool for third-party verification and validation, as well as for future development of other higher-order continuum damage models.

An adaptive global–local approach for phase-field modeling of anisotropic brittle fracture

Article

Full-text available

Nov 2019
COMPUT METHOD APPL M

This work addresses an efficient Global–Local approach supplemented with predictor–corrector adaptivity applied to anisotropic phase-field brittle fracture. The phase-field formulation is used to resolve the sharp crack surface topology on the anisotropic/non-uniform local state in the regularized concept. To resolve the crack phase-field by a given single preferred direction, second-order structural tensors are imposed to both the bulk and crack surface density functions Accordingly, a split in tension and compression modes in anisotropic materials is considered. A Global–Local formulation is proposed, in which the full displacement/phase-field problem is solved on a lower (local) scale, while dealing with a purely linear elastic problem on an upper (global) scale. Robin-type boundary conditions are introduced to relax the stiff local response at the global scale and enhancing its stabilization. Another important aspect of this contribution is the development of an adaptive Global–Local approach, where a predictor–corrector scheme is designed in which the local domains are dynamically updated during the computation. To cope with different finite element discretizations at the interface between the two nested scales, a non-matching dual mortar method is formulated. Hence, more regularity is achieved on the interface. Several numerical results substantiate our developments.

A phase-field regularized cohesive zone model for hydrogen assisted cracking

Article

Full-text available

Aug 2019
COMPUT METHOD APPL M

Being able to seamlessly deal with complex crack patterns like branching, merging and even fragmentation, phase-field fracture/damage models are promising in the modeling of localized failure in solids. This paper addresses a phase-field regularized cohesive zone model (PF-CZM) for hydrogen assisted cracking based on our previous work on purely mechanical problems. Two distinct hydrogen enhanced decohesion mechanisms are dealt with by introducing various implicitly defined (via the crack phase-field) hydrogen-dependent softening laws. The presented model are then numerically tested and compared against several benchmark examples. It is found that, though the PF-CZM gives different results regarding various decohesion mechanisms, the global responses are insensitive to both the mesh discretization resolution and the incorporated length scale parameter even in the presence of hydrogen.

Circumventing mesh bias by r-and h-adaptive techniques for variational eigenfracture

Article

Full-text available

Dec 2019
INT J FRACTURE

This article introduces and compares mesh r-and h-adaptivity for the eigenfrac-ure model originally proposed in [1, 2], with the goal of suppressing potential mesh bias due to the element deletion. In the r-adaptive approach, we compute the configurational force at each incremental step and move nodes near the crack tip parallel to the normalized configurational forces field such that the crack propagation direction can be captured more accurately within each incremental step. In the h-adaptive approach, we introduce mesh refinement via a quad-tree algorithm to introduce more degrees of freedom within the nonlocal-neighborhood such that a more refined crack path can be reproduced with a higher mesh resolution. Our numerical examples indicate that the r-adaptive approach is able to replicate curved cracks and complex geometrical features, whereas the h-adaptive approach is advantageous in simulating sub-scale fracture when the nonlocal regions are smaller than the unrefined coarse mesh.

Meta-modeling game for deriving theory-consistent, microstructure-based traction-separation laws via deep reinforcement learning

Article

Full-text available

Feb 2019
COMPUT METHOD APPL M

This paper presents a new meta-modeling framework to employ deep reinforcement learning (DRL) to generate mechanical constitutive models for interfaces. The constitutive models are conceptualized as information flow in directed graphs. The process of writing constitutive models is simplified 8 as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). Thus meta-modeling can be formulated as a Markov decision process with well-defined states, actions, rules, objective functions and rewards. By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive model it generated through self-playing, in the same way AlphaGo Zero (the algorithm that outplayed the world champion in the game of Go) improves its gameplay. Our numerical examples show that this automated meta-modeling framework not only produces models which outperform existing cohesive models on benchmark traction-separation data, but is also capable of detecting hidden mechanisms among micro-structural features and incorporating them in constitutive models to improve the forward prediction accuracy, which are difficult tasks to do manually.

An updated Lagrangian LBM-DEM-FEM coupling model for dual-permeability fissured porous media with embedded discontinuities

Article

Full-text available

Sep 2018
COMPUT METHOD APPL M

Many engineering applications and geological processes involve embedded discontinuities in 6 porous media across multiple length scales (e.g. rock joints, grain boundaries, deformation bands and 7 faults). Understanding the multiscale path-dependent hydro-mechanical responses of these interfaces across 8 length scales is of ultimate importance for applications such as CO 2 sequestration, hydraulic fracture and 9 earthquake rupture dynamics. While there exist mathematical frameworks such as extended finite element 10 and assumed strain to replicate the kinematics of the interfaces, modeling the cyclic hydro-mechanical 11 constitutive responses of the interfaces remains a difficult task. This paper presents a semi-data-driven 12 multiscale approach that obtains both the traction-separation law and the aperture-porosity-permeability 13 relation from micro-mechanical simulations performed on representative elementary volumes in the finite 14 deformation range. To speed up the multiscale simulations, the incremental constitutive updates of the 15 mechanical responses are obtained from discrete element simulations at the representative elementary vol-16 ume whereas the hydraulic responses are generated from a neural network trained with data from lattice 17 Boltzmann simulations. These responses are then linked to a macroscopic dual-permeability model. This 18 approach allows one to bypass the need of deriving multi-physical phenomenological laws for complex 19 loading paths. More importantly, it enables the capturing of the evolving anisotropy of the permeabilities 20 of the macro-and micro-pores. A set of numerical experiments are used to demonstrate the robustness of 21 the proposed model. 22

On the crack-driving force of phase-field models in linearized and finite elasticity

Article

May 2019
COMPUT METHOD APPL M

The phase-field approach to fracture has been proven to be a mathematically sound and easy to implement method for computing crack propagation with arbitrary crack paths. Hereby crack growth is driven by energy minimization resulting in a variational crack-driving force. The definition of this force out of a tension-related energy functional does, however, not always agree with the established failure criteria of fracture mechanics. In this work different variational formulations for linear and finite elastic materials are discussed and ad-hoc driving forces are presented which are motivated by general fracture mechanical considerations. The superiority of the generalized approach is demonstrated by a series of numerical examples.

A phase-field formulation for dynamic cohesive fracture

Article

Feb 2019
COMPUT METHOD APPL M

We extend a phase-field/gradient damage formulation for cohesive fracture to the dynamic case. The model is characterized by a regularized fracture energy that is linear in the damage field, as well as non-polynomial degradation functions. Two categories of degradation functions are examined, and a process to derive a given degradation function based on a local stress–strain response in the cohesive zone is presented. The resulting model is characterized by a linear elastic regime prior to the onset of damage, and controlled strain-softening thereafter. The governing equations are derived according to macro- and microforce balance theories, naturally accounting for the irreversible nature of the fracture process by introducing suitable constraints for the kinetics of the underlying microstructural changes. The model is complemented by an efficient staggered solution scheme based on an augmented Lagrangian method. Numerical examples demonstrate that the proposed model is a robust and effective method for simulating cohesive crack propagation, with particular emphasis on dynamic fracture.

A Variational Phase-Field Model for Hydraulic Fracturing in Porous Media

Article

Apr 2019
COMPUT METHOD APPL M

Rigorous coupling of fracture-porous medium fluid flow and topologically complex fracture propagation is of great scientific interest in geotechnical and biomechanical applications. In this paper, we derive a unified fracture-porous medium hydraulic fracturing model, leveraging the inherent ability of the variational phase-field approach to fracture to handle multiple cracks interacting and evolving along complex yet, critically, unspecified paths. The fundamental principle driving the crack evolution is an energetic criterion derived from Griffith’s theory. The originality of this approach is that the crack path itself is derived from energy minimization instead of additional branching criterion. The numerical implementation is based on a regularization approach similar to a phase-field model, where the cracks location is represented by a smooth function defined on a fixed mesh. The derived model shows how the smooth fracture field can be used to model fluid flow in a fractured porous medium. We verify the proposed approach in a simple idealized scenario where closed form solutions exist in the literature. We then demonstrate the new method’s capabilities in more realistic situations where multiple fractures turn, interact, and in some cases, merge with other fractures.

A phase field model for cohesive fracture in micropolar continua

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