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under the supervision of

Chair for Computation in Engineering

Technische Universitat Munchen

Phase-field modeling of brittlefracture with multi-level hp-FEM

and the Finite Cell Method

Master’s thesis

for the Master of Science programme Computational Sciencesin Engineering

Sindhu Nagaraja

Examiners: Prof. Dr.-Ing. Laura De Lorenzis

Prof. Dr.-Ing. habil. Manfred Krafczyk

Prof. Dr. rer. nat Ernst Rank

Supervisors: Mohamed Elhaddad, M.Sc.

Advisors: Dr.-Ing Stefan Kollmannsberger

Dr.-Ing Marreddy Ambati

Date of issue: 03. April, 2017

Date of submission: 29. September, 2017

2

Involved Organisations

Chair for Computation in Engineering

Department of Civil, Geo and Environmental Engineering

Technische Universitat Munchen

Arcisstraße 21

D-80333 Munchen

Institute of Applied Mechanics

Department of Civil Engineering and Environmental Science

Technische Universitat Braunschweig

Pockelsstraße 3 (Okerhochhaus)

38106 Braunschweig

4

Tasks

Implementation of a phase-field model for brittle fracture intoAdhoC++, the hp-

FEM and Finite Cell Code of the Chair for Computation in Engineering in two

dimensions.

Verification of the implementation on benchmark examples with homogeneous

material.

Examples for non-geometry conforming mesh using FCM.

Development of a heuristic refinement criterion.

Extension of the method for adaptively refined meshes.

Comparative studies for uniform and adaptively refined meshes.

Acknowledgements

This Master’s thesis has been written by me in September 2017 to fulfill one of the

requirements of my Graduate Program, Master of Science - Computational Sciences in

Engineering (CSE). The work has been carried out at the Chair for Computation in

Engineering, Technische Universitat Munchen. I would like to express my gratitude

towards Prof. Dr.-Ing. Laura De Lorenzis and Prof. Dr. rer. nat. Ernst Rank for

providing me this opportunity which broadened my research perspective.

This work is partly based on the research carried out by Dr.-Ing. Marreddy Ambati. I

would like to thank him for sharing his work. I am greatful for the guidance provided by

Dr.-Ing Stefan Kollmannsberger throughout my thesis. His cooperation, motivational

words and timely input played an important role in this work. I earnestly thank

Mohamed Elhaddad for mentoring me and supporting me at every stage, without which

it would have been impossible to accomplish this work. Furthermore, special thanks

to Phillip Kopp for his cooperation and inputs during the thesis. I also acknowledge

the hospitality of the members of the Chair of Computation in Engineering and thank

them for hosting me during the thesis.

Munchen, September 2017 Sindhu Nagaraja

Abstract

Phase-field approach for computational fracture mechanics is an elegant numerical

technique to predict fracture based failures in materials and components employed in

various engineering applications.

The multi-level hp-FEM has proven to yield an exponential convergence of the approx-

imation error even for problems with singular solution characteristics. This method

also enables a dynamically changing mesh which allows the refinement to stay local

near singularities or high gradients. This feature of the multi-level hp-FEM is ideally

suited to track sharp features such as propagation of cracks.

The ability to accurately predict fracture in industrial applications involving complex

geometries and loading conditions is gaining importance like never before in the

past decade. To achieve this, the conventional Finite Element Method considers a

mesh whose boundaries have to coincide with the boundaries of the geometry under

consideration. This makes the mesh generation process tedious. The Finite Cell

Method based on higher order elements, provides an alternate mesh generation process

and is an embedded domain method.

The research presented in this work focuses on integrating a two-dimensional phase-

field framework for both quasi-static and dynamic brittle fracture developed at the

Institut fur Angewandte Mechanik, TU Braunschweig with the multi-level dynamically

adaptive hp-framework and the Finite Cell Method developed at the Lehrstuhl fur

Computation in Engineering, TU Munchen. The key objective is to implement a phase-

field model into Adhoc++, the hp-FEM and the Finite Cell code of the Lehrstuhl fur

Computation in Engineering at TU Munchen and to develop a refinement criterion

that ensures a dynamically changing mesh which is adaptive in nature. The numerical

results presented in the thesis illustrate the potential of the application of the uniform

multi-level hp-refinement and the FCM in the context of phase-field models for both

quasi-static and dynamic brittle fracture.

Contents

List of Figures 2

List of Tables 4

List of Symbols 6

1 Introduction 8

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Fundamental concepts 12

2.1 Basics of fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Modes of fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Linear elastic fracture mechanics (LEFM) . . . . . . . . . . . . 13

2.1.3 Griffith’s criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Phase field approximation of crack topology . . . . . . . . . . . . . . . 15

2.3 Multi-level hp-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 The Finite Cell Method . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Basic concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.2 Application of non-boundary conforming boundary conditions . 21

3 Formulation 23

3.1 Quasi-static brittle fracture . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Extension to dynamic brittle fracture . . . . . . . . . . . . . . . . . . . 27

3.3 Numerical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.1 Quasi-static brittle fracture . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Dynamic brittle fracture . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Finite Cell Method for phase-field quasi-static brittle fracture . . . . . 32

4 Numerical results 34

4.1 Single-edge notched tension test . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Validation of implementation of the hybrid formulation . . . . . 35

4.1.2 Parametric influence of convergence behavior of the phase-field

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Single-edge notched shear test . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Validation of implementation of the anisotropic Miehe formulation 42

4.2.2 Parametric influence of convergence behavior of the phase-field

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.3 Comparative study between uniform and adaptively refined meshes 48

1

Contents

4.3 Notched plate with hole . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Parametric influence of number of staggered iterations on crack

propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Crack propagation under compressive loads . . . . . . . . . . . . . . . . 54

4.5 Dynamic crack branching . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Summary and outlook 59

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Bibliography 61

2

List of Figures

1.1 Examples for fracture induced engineering failures . . . . . . . . . . . . 8

2.1 Modes of Fracture [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Phase-field approximation. a) Sharp crack topology in the domain of

the solid. b) Regularized crack surface using length scale parameter l . 16

2.3 One-dimensional shape functions . . . . . . . . . . . . . . . . . . . . . 18

2.4 Two-dimensional mode categories . . . . . . . . . . . . . . . . . . . . . 18

2.5 Refinement by superposition [10], [11] . . . . . . . . . . . . . . . . . . . 18

2.6 The multi-level hp-FEM concept [11] . . . . . . . . . . . . . . . . . . . 19

2.7 Basic concept of the Finite Cell Method [19] . . . . . . . . . . . . . . . 20

2.8 Quadtree integration [19] . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Schematic representation of the domain Ω with internal discontinuity Γ . 23

4.1 Geometry and boundary conditions for single-edge notched tension test 35

4.2 Single-edge notched tension test. Validation of crack evolution with

hybrid formulation against reference results. . . . . . . . . . . . . . . . 36

4.3 Single-edge notched tension test. Validation of load-displacement be-

havior with hybrid formulation against reference results. . . . . . . . . 36

4.4 Single-edge notched tension test. Multi-level hp-refinement for different

ansatz orders with k = 6. Crack phase-field at different displacements. 38


ansatz orders with k = 6. Load-displacement curves. . . . . . . . . . . 39


refinement depths with p = 3. Crack phase-field evolution at different

displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


refinement depths with p = 3. Load-displacement curves. . . . . . . . . 41

4.8 Geometry and boundary conditions of single-edge notched shear test . . 42

4.9 Single-edge notched shear test. Validation of crack evolution with the

anisotropic Miehe formulation against reference results. . . . . . . . . . 43

4.10 Single-edge notched shear test. Validation of load-displacement behavior

with the anisotropic Miehe formulation against reference results. . . . . 43

4.11 Single-edge notched shear test. Multi-level hp-refinement for different

ansatz orders with k = 6. Crack phase-field at different displacements. 45


ansatz orders with k = 6. Load-displacement curves. . . . . . . . . . . 46


refinement depths with p = 5. Crack phase-field at different displacements. 47

3

List of Figures


refinement depths with p = 5. Load-displacement curves. . . . . . . . . 48

4.15 Single-edge notched shear test. Comparison of crack phase-field for

uniformly and adaptively refined meshes with p = 1 at different dis-

placements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.16 Single-edge notched shear test. Comparison of load-displacement curves

for uniformly and adaptively refined meshes with p = 1. . . . . . . . . . 50

4.17 Geometry and boundary conditions of notched plate with hole . . . . . 51

4.18 Notched plate with hole. Crack pattern from the experiment . . . . . . 51

4.19 Notched plate with hole. Multi-level hp-refinement: p = 2, k = 4. Crack

phase-field at different displacements. . . . . . . . . . . . . . . . . . . . 52

4.20 Notched plate with hole. Multi-level hp-refinement for different number

of staggered iterations : p = 3, k = 4. Crack phase-field at different

displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.21 Geometry and boundary conditions of concrete block under compression. 54

4.22 Concrete block under compression. Crack pattern from the experiments

carried out by Dipl.-Ing. Gerald Schmidt-Thro, Lehrstuhl fur Massivbau,

TU Munchen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.23 Concrete block under compression. Multi-level hp-refinement: p = 1, k

= 4. Crack phase-field at different displacements. . . . . . . . . . . . . 55

4.24 Geometry and loading of dynamic uniaxial tension test. . . . . . . . . . 56

4.25 Dynamic crack branching under uniaxial tension. Multi-level hp-

refinement for different ansatz order with k = 5 using the anisotropic

Miehe formulation. Crack phase-field evolution over time. . . . . . . . . 57


refinement for different ansatz order with k = 5. Plot of strain energy

overtime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


refinement for different ansatz order with k = 5 using the hybrid

formulation. Crack phase-field evolution over time. . . . . . . . . . . . 58

4

List of Tables

4.1 Single-edge notched shear test. Parametric comparison for uniformly

and adaptively refined meshes with p = 1 and the hybrid formulation. . 50

5

List of Symbols

Abbreviations

Abbreviation Stands for

FEM Finite Element Methods

FCM Finite Cell Method

PDE Partial Differential Equation

SIF Stress Intensity Factor

LEFM Linear Elastic Fracture Mechanics

EPFM Elasto Plastic Fracture Mechanics

GP Gauß Points

DOFs Degrees Of Freedom

Symbols used in Formulae

Latin

Symbol Stands for Unit

l length scale parameter [m]

s phase field or crack field paramter [ ]

a crack length [m]

u displacement field [m]

L Lagrangian [J]

H history field [ Jm2 ]

E material stiffness [Pa]

Gc fracture toughness [Nm ]

h grid size in the square mesh [m]

p ansatz order of shape functions [ ]

k refinement depth in multi-level hp-refinement [ ]

∆u displacement increment [m]

dt temporal discretization or time step [s]

6

List of Tables

Greek

Symbol Stands for Unit

Ω the three dimensional domain (body) [m3]

Ωphy the three dimensional physical domain in the

context of the FCM

[m3]

Ωfict the three dimensional fictitious domain in the

context of the FCM

[m3]

Ω∪ union of the physical and the fictitious domain

in the context of the FCM

[m3]

Γ crack boundary [m2]

Ψkin kinetic energy [J ]

Ψpot potential energy [J ]

Ψe elastic strain energy density [ Jm3 ]

ε strain tensor [ ]

σ strain tensor [Pa]

λ Lame’s first parameter [Pa]

µ Lame’s second parameter [Pa]

η numerical paramter to model artificial stiffness [ ]

ρ mass density [ kgm3 ]

ν poisson’s ratio [ ]

α indicator function in the context of the FCM [ ]

7

1 Introduction

1.1 Motivation

Damage-based design of engineering structures has become one of the most important

design strategies in the past few decades. This ”prevention is better than cure” design

principle focuses mainly on inhibiting fracture-induced failure. The dramatic failure of

several Liberty ships during the World War II was a consequence of brittle fracture in

hull and decks of the ships that were made up of steel, see Fig:1.1a. The derailment

of the train ICE 884 at Eschede, Germany killing 101 people, depicted in Fig:1.1b,

is owed to the fatigue fracture of the wheel rim [3]. Several engineering structures

used in oil plants, reservoir construction, nuclear power plants are sensitive to the

presence of cracks. Since it is almost impossible to have a perfect crack free material,

it becomes necessary to study the performance and durability of such structures under

the given loading with a pre-existing crack. These serve as motivating examples for the

development of efficient fracture models and numerical simulation of fracture processes.

In this regard, a varitey of fracture models have been introduced in the research

community dedicated to damage mechanics. The most successful theory proposed for

brittle fracture is an energy based theory, proposed by Griffith. It relates the dissipated

energy during crack propagation to the energy required to form new crack surfaces

and is briefly introduced in section 2.1.2.

(a) Failure of Liberty ship S.S.Schenectady,1943 [1]

(b) ICE accident due to fracture ofwheel on June 3, 1998 [2]

Figure 1.1: Examples for fracture induced engineering failures

Most of the engineering problems are now solved numerically. Galerkin based Finite

Element Method (FEM) has become the most admired numerical methods to solve

partial differential equations (PDEs) and is undoubtedly the first choice of today’s

8

1 Introduction

engineers. One of the many complexities involved in fracture processess is to model the

crack. A crack is discontinuous by its very nature and this poses a challenge in modeling

a crack in finite element setting. Furthermore, it is necessary to track the crack as it

propagates. In the research dedicated to fracture mechanics, Extended Finite Element

Methods (XFEM), Cohesive Zone Methods, Cohesive Segment Methods, Discontinuous

Galerkin Method (DG), Phase-Field Modeling are a few popular computational meth-

ods. In this thesis, we study the phase-field approach for quasi-static and dynamic

brittle fracture. The phase-field method eliminates the necessity to algorithmically

track the discontinuous displacements across the crack path by modeling the discrete

crack as a smooth continuous function that elegantly transits from an undamaged

region to a fully cracked region.

Although FEM is the most sought after method to simulate structural engineering

problems, there is always a constant demand for simulating problems with higher

complexities and intricacies while using minimum resources and time with higher

accuracy in the numerical solution. This can be achieved by refinement techniques.

The most trivial one is the uniform h-refinement wherein the solution accuracy is

increased by increasing the number of elements. To this end, elements are bisected

in each direction, yielding four new elements. This is continued until convergence is

reached. Although this has been proven to yield good results, the rate at which the error

decreases is rather slow. For extremely accurate results, a very large number of degrees

of freedom might become necessary and hence a pure uniform h-refinement becomes

computationally expensive. Another method that has proven to yield exponential

convergence for smooth problems is the p-refinement technique. Here, higher order

shape functions are used which approximates the analytical solution better using lesser

number of degrees of freedom, compared to h-FEM. In the recent past, non-uniform

rational B-splines (NURBS) are being used for even better results and for higher

continuity in the solution in the context of Isogeometric Analysis (IGA).

The triumph of the above mentioned p-refinement technique starts deteriorating to an

algebraic convergence in the presence of discontinuities, singularities or high gradients

in the solution. The subject of this thesis has a discontinuity smoothened by the

phase-field function. Although this function is continuous, it has high gradients. This

is the trigerring point to use a refinement strategy in the study which yields excellent

convergence even in the presence of high gradients. A felicitous refinement technique

for this situation is the hp-FEM. In this refinement version, the advantages of both

h-version and p-version are exploited. While the elements at the vicinity of high

gradients are refined and lower order polynomials are used for solution approximation,

elements away from the sensitive zone remain large and higher order polynomials

are used to smoothly approximate the solution. This improves the solution accuracy

significantly, compared to h-FEM and p-FEM with comparable number of unknowns.

It is evident from the numerical analysis presented in Chapter 4 that the strong

gradients in the phase-field can be used to drive a dynamic refinement (a mesh that

changes over the course of the simulation) thereby recovering excellent convergence.

9

1 Introduction

While the convergence behavior of the hp-version is promising, it has not yet become

the most popular method in engineering applications owing to the challenges it poses,

the most challenging aspect being the hanging nodes generated in a local h-refinement.

These hanging nodes are to be constrained appropriately to yield atleast C0 continutiy

in the numerical solution. This process is tedious and algorithmically challenging.

Alternatively, hanging nodes which result in an irregular unstructured mesh can be

eliminated by appropriate refinement of the adjacent elements thus resulting in a

regular mesh. However, this method conflicts the idea of local refinement.

Another implementation of the hp-version of FEM adopts a refinement by superposition

rather than a refinement by replacement as described above. This is called the multi-

level hp-scheme. The idea of this is to superpose a base mesh with a finer overlay

mesh in such locations of the domain where there are singularities or high gradients

in the solution. The fundamental principle of this method to acheive inter-element

continuity is to apply homogeneous Dirichlet boundary conditions on the boundary

of the refinement zone. This method has proven to bring down the implementational

complexity related to hanging nodes while constructing the same Finite Element space

and thereby retaining the same approximation quality as that from a conventional

hp-FEM method. This refinement technique is used throughout in this work.

The Finite Cell Method (FCM) is an embedded or immersed boundary method which

is based on high order Finite Elements. It is a non-geometry conforming discretization

technique which does not resolve the geometry explicitly, rather embedds the physical

domain into a larger ficticious domain. This larger embedding domain is of simple

geometry and hence can be easily discretized thus evading the numerically expensive

and time consuming meshing procedure. The original geometry is recovered during

integration of the weak form by imposing a penalty on the integration points that

do not belong to the actual geometry. This idea not only enables complex geometry

representations but also is handy to use when the geometry under consideration is

changing with time. Section 4.3 considers an example of a notched plate with hole,

where the FCM has been applied successfully in the context of phase-field models.

1.2 Outline of the thesis

This thesis is organised into four chapters. The first chapter as already seen motivates

the under taken study. The second chapter introduces the concepts used in the thesis.

Groundwork required to understand a fracture problem, phase-field models, higher

order FEM, multi-level hp-FEM and FCM are laid here. The third chapter presents

the governing equations of a phase-field based brittle fracture model followed by its

numerical formulation in both, FEM and FCM. Firstly, the quasi-static model is

presented which is later extended to a dynamic model. Two different phase-field

formulations are used in this thesis, anisotropic Miehe and hybrid formulation. The

10

1 Introduction

last chapter is dedicated to the application of phase-field model, multi-level hp-FEM

and FCM to benchmark problems. A plane strain problem in which a plate with a

pre-existing crack subjected to two different loading conditions is simulated using multi-

level hp-FEM. This is followed by an example involving a non-geometry conforming

mesh solved with the FCM. To observe the behavior of Miehe and Hybrid formulations

for compression, a plane strain problem where a concrete block with no pre-existing

crack subjected to compressive loading is discussed. Finally, an attempt is made to

extend the implementation to a dynamic crack branching problem in which a plane

strain example under uniaxial tension is presented.

11

2 Fundamental concepts

2.1 Basics of fracture mechanics

Fracture Mechanics deals with the phenomenon of crack initiation, crack growth and

its propagation. These phenomena are sensitive and are influenced by various factors.

For instance, the process of fracture is highly dependent on the nature of the material.

Hence, there are different theories adopted for different materials. Brittle materials

are those which exhibit poor resistance to the external tensile loading. They undergo

failure without exhibiting significant deformations. Brittle fracture involves almost nil

or extremely small plastic deformation zones when compared to ductile materials and

is generally instantaneous. Hence, we may conclude that it is necessary to develop two

distinct models in order to represent features of ductile and brittle fracture. While

Linear Elastic Fracture Mechanics (LEFM) is suitable mostly for brittle materials since

they have very small plastic zone and hence have elastic response to the external applied

load, Elasto-Plastic Fracture Mechanics (EPFM) is to be used for materials that exhibit

evident plasticity. In addition to the material behavior, it is necessary to consider the

nature of the loading the component is subjected to. For example, dynamic problems,

creep, fatigue, etc are to be treated individually with suitable theories. Therefore,

given a load, material, geometry and a pre-exisiting crack, fracture mechanics helps to

estimate the maximum load a component can withstand before failure due to crack

propagation, also answering questions such as: Will the existing crack grow under the

applied external load? If it grows, with what speed does it? Will the crack growth be

arrested due to dissipative effects? When will the component fail?

2.1.1 Modes of fracture

External loads can be classified into three independent types, thus, leading to a

simplified scenario where the effect of each type can be determined individually. The

principle of superposition can then be applied in an appropriate manner to find the

combined effect of different loads. The three independent modes into which the loading

can be split are:

MODE 1 - Opening Mode - This mode arises due to a tensile loading perpendicular

to crack. It is the most common mode of fracture and also exhibits the highest

Stress Intensity Factor (SIF).

MODE 2 - Shearing Mode - Here, applied stress is shear in the in-plane direction

of the crack. In other words, applied stress is perpendicular to leading edge of

12


the crack but in the plane of the crack.

MODE 3 - Tearing or Twisting Mode - Here, the applied external stress is out

of plane, i.e., the stress is parallel to the leading edge of the crack.

Figure 2.1: Modes of Fracture [4]

2.1.2 Linear elastic fracture mechanics (LEFM)

In fracture problems, the so-called process zone is the zone in which atomic bond

breaking occurs, thereby, leading to crack propagation. The size of this zone determines

the facet of the fracture process. Linear Elastic Fracture Mechanics deals with those

problems in which the process zone (also called plastic zone) is negligibly small. It

is assumed that the plastic zone around the crack tip is negligible when compared

to all other macroscopic dimensions of the physical domain. This is true for most

brittle materials. The other assumption of LEFM is that the material response is

linearly elastic throughout the fracture process. However, this assumption leads to

singularities in stresses at crack tip. (infinite stress at crack tip). In reality, the plastic

zone size is never a perfect zero and hence infinite stresses do not occur at the crack

tip. As long as this yielding zone is negligibly small, linear theory is still an acceptable

assumption. Hence, it is only logical to apply LEFM for brittle fracture, which is why

the formulation presented in this work assumes LEFM.

The characterizing parameter of LEFM is the Stress Intensity Factor. The SIF serves

as a crack propagation criteria for brittle fracture or in general LEFM. If the value of

the SIF at crack tip exceeds a critical SIF, then the crack propagates:

KI ≥ KIC (2.1)

13


2.1.3 Griffith’s criteria

Griffith’s criteria is based on an energy balance. It is applicable for both elastic and

plastic cases. The fracture surface energy Γs is the energy necessary to create the

newly formed cracked surfaces during crack propagation. This energy is assumed to

be proportional to the area of the crack.

Γs = GcA (2.2)

The material constant Gc is called crack resistance, crack resistance force or fracture

toughness. It is the energy required to create a unit area of fracture surface. The

energy balance states that,

E + K + Γs = W +Q (2.3)

where E is the internal energy, K is the kinetic energy, W is the rate of work associated

to the external forces and Q is the rate of heat supply.

Assuming that no heat transfer occurs, we have

Γs = W − E − K (2.4)

If Γs ≥ 0 crack starts to propagate.

Now the quasi-static case is considered, where the contribution of kinetic energy is

negligible. Thus, the energy balance equation becomes

Γs = W − E (2.5)

Assuming a pure elastic case, the internal energy E and the external work W can

be expressed in terms of potentials. They take the forms E = Πint and W = −dΠext

dt .

Thus, the above equation simplifies to

dΠ

dt+dΓsdt

= 0 (2.6)

where Π = Πint + Πext is the total potential.

For a line crack of length a, the above further simplifies to[dΠ

da+dΓsda

]da

dt= 0 (2.7)

The energy release rate G is defined as G = −dΠda and the critical energy release rate

(or fracture toughness) Gc = −dΓs

da which leads to the final crack propagation criteria

as obtained using Griffith’s theory

(Gc −G)a = 0 (2.8)

Thus, we have G = Gc for initiation of crack propagation.

14


2.2 Phase field approximation of crack topology

Fracture problems are discontinuous by nature. Methods such as XFEM, Cohesive Zone

Methods, Cohesive Segment Methods incorporate this discontinuity in the primary

field variable which is the displacement field. These are called discrete fracture models

since they represent the crack as a discontinuous entity. They make use of remeshing

strategies or enrichment of displacement fields. These methods involve algorithmically

tracking the crack during the course of the simulation which is complicated if the

crack geometry is intricate. Although these models represent crack propagation well

in two dimensions, it is numerically expensive and challenging to extend these in three

dimensions.

To overcome the difficulty of tracking crack surfaces, a new strategy called phase-

field modeling was developed in the late 1990’s. In this method, no discontinuity is

introduced to the displacement variable, instead a new field variable also called the field

order parameter, is introduced which ensures a smooth transition between discontinuity

and continuity. A phase-field model can model systems with sharp interfaces. In

fracture mechanics perspective, this field order parameter is called the crack field or

phase field and throughout this script it is denoted by s. This crack field is continuous

but is a scalar field variable. It describes a smooth transition between the undamaged

material and fully cracked material. This method is based on the minimization of the

total energy of the system with respect to the displacement field and the crack field.

s =

0 : Fully cracked material

1 : Undamaged material(2.9)

The evolution of this crack field under the given loading represents the process of

fracture. While an a-priori knowledge of the crack path becomes essential in discrete

fracture models, this becomes unnecessary for phase-field models since the evolution

of crack field represents the crack surface. Processes like crack propagation, kinking,

crack branching which involve complex crack surfaces are relatively simple to simulate

using phase-field models.

Mathematically, phase-field models are two coupled non-linear system of Partial

Differential Equations. The first equation is the balance equation obtained from

continuum mechanics and second one is the partial differential equation that describes

the evolution of the phase-field parameter s. Solutions to these coupled equations can

be obtained in two ways, using a monolithic scheme or a non-Monolithic or staggered

scheme. In the monolithic scheme, the two equations are solved simultaneously to

obtain solutions for the displacement field and the crack phase-field. In the staggered

scheme, first the displacement equation is solved for, assuming crack field to be constant

15


and then using the obtained displacement field, the phase field equation is solved. The

obtained phase field solution is then used to solve again for displacement field. This

iterative process is continued until a certain accuracy is reached.

Yet another important aspect of phase-field models is the so-called length scale parame-

ter, denoted by l. This parameter controls the width of transition from the undamaged

to the fully cracked zone and can be interpreted as the width of the regularized crack,

see Fig:2.2. Phase-field models converge to Griffith’s theory for fracture of brittle

materials if the value of length scale parameter tends to zero [5]. On the outset, it

is appealing to set a very low value of l. But for small length scale parameter, the

spatial and in-turn the temporal discretization in the case of dynamic fracture, should

be fine which leads to a huge stiffness matrix thereby increasing the computational

cost. Recent advances in high performance computing (HPC) suggest, however, to

exploit parallel computing for solving problems on very fine discretizations that can

capture very small values of l [17].

Γ

l s = 0

s = 1

Ω

∂Ωgi

∂Ωhi

Γ

l

Ω

∂Ωgi

∂Ωhi

Figure 2.2: Phase-field approximation. a) Sharp crack topology in the domain of the

solid. b) Regularized crack surface using length scale parameter l

In the research dedicated to phase-field approach for fracture problems, there are

several models proposed by the mechanics community and also by the physicists, for

details refer [5] . In the scope of this work, followed is the regularized variational

formulation presented by Miehe et al [7] and the recently proposed hybrid formulation

by Ambati et al [5]. Both the formulations are presented in the next chapter.

2.3 Multi-level hp-FEM

In this section, the basic concept of the multi-level hp-FEM is introduced. As mentioned

earlier, this refinement technique combines the advantages of both h-version and p-

version ensuring exponential convergence even in the presence of high gradients or

16


singularities in the solution field which is the case in fracture problems. While classical

FEM is generally implemented using Lagrange shape funtions which by nature are

nodal basis functions as shown in the figure Fig:2.3(a), the p-version makes use of

integrated Legendre polynomials. This is because, the condition number of the stiffness

matrix using Largrange basis funtions increase drastically for higher polynomial orders

making them unsuitable for p-version. Thus, the p-version as introduced by Babuska

et al [8] makes use of integrated Legendre polynomials such that it yields excellent

properties of the stiffness matrix. The Legendre polynomials have a unique property

that they are orthogonal. Since the stiffness matrix uses the derivatives of shape

functions, this orthogonality property needs to be transferred to the first derivative of

the shape functions. Integrated Legendre polynomials serve this purpose. All shape

functions except the linear ones are orthogonal to each other. As a result, the stiffness

matrices for 1D linear elastic problems are almost diagonal except the product of

the linear modes, thus improving its condition number. However, this orthogonolity

property does not hold in multiple dimensions. Despite this, the condition number

of the stiffness matrices using an Integrated Legendre basis are significantly better

compared to that using a Lagrange basis. Furthermore, the integrated Legendre shape

functions are hierarchical in nature. That implies, there is an addition of one shape

function for every order of the polynomial that is increased while all other shape

functions remain same. The one-dimensional integrated Legendre polynomials are

depicted in Fig:2.3(b) and are defined as,

N1(ξ) =1

2(1 + ξ)

N2(ξ) =1

2(1− ξ)

Ni(ξ) = Pi−1(ξ) i = 3, 4, ..., p+ 1

(2.10)

where, p is the polynomial order and Pi is defined in terms of the Legendre polynomials

Li:

Pi(ξ) =1√

4i− 2(Li(ξ)− Li−2(ξ)) i = 2, 3, 4, ... (2.11)

A tensor product of one dimensional basis functions is used to obtain higher dimensional

shape functions, see Eq:2.12. Following Zander et al [9], the two-dimensional tensor

product can be classified into three categories which plays an important role in the

definition of C0-continuous global basis functions: nodal modes, edge modes and

bubble modes, as depicted in Fig:2.4.

N2Di,j (ξ, η) = N1D

i (ξ)N1Dj (η)

N3Di,j,k(ξ, η, µ) = N2D

i,j (ξ, η)N1Dk (µ)

(2.12)

17


(a)Lagrange shape functions (b) Integrated Legendre shape functions

Figure 2.3: One-dimensional shape functions

(a) Nodal mode (b) Edge mode (c) Bubble mode

Figure 2.4: Two-dimensional mode categories

Having had an overview of the hierarchical shape functions which forms an essential

ingredient of multi-level hp-FEM, the basic outline of multi-level hp-FEM is presented

here. The core idea of this refinement technique is to enhance the quality of the

solution by superposing finer overlay elements over those base elements which are to

be refined. The final approximation of the solution u is then the sum of base mesh

solution ub and the overlay solution u0 as illustrated in Fig:2.5. This enables an

optimal refinement using finer elements with low polynomial order in the vicinity of

singularities and strong gradients to ensure small discretization errors and a better

quality of solution while still using larger elements with higher polynomial orders where

the solution is smooth.

Base mesh solution ub Overlay solution u0 Final solution u

Figure 2.5: Refinement by superposition [10], [11]

18


Unlike classical refine by replacement hp-refinement version, the multi-level hp-FEM

evades the algorithmic implementational difficulties by imposing two simple require-

ments for convergence. Namely, the compatibility and linear independence of basis

functions. Compatibility is achieved by imposing homogeneous Dirichlet boundary

conditions on each layer of superposed overlay mesh, thus maintaining C0 continuity.

This corresponds to deactivating those components which are connected to the bound-

ary of the overlay mesh. In one dimension, nodes, in two dimensions nodes and edges,

and in three dimensions nodes, edges and faces on the overlay boundary needs to be

deactivated. Linear independence of basis is achieved by deactivating all topological

components that have active sub-components. That is, high-order shape functions are

deactivated on those elements which have an underlying child element. This concept is

explained in Fig:2.6 where k is the refinement depth and p is the ansatz order. These

two impositions eliminate the burden of algorithmic constraining of hanging nodes.

Figure 2.6: The multi-level hp-FEM concept [11]

2.4 The Finite Cell Method

One of the fundamental ideas of the Finite Element Method is the isoparametric

mapping where same discretization is used to represent the solution field of the

PDE under consideration as well as the geometry. This imposes a requirement of

”good” elements to ensure adequate numerical accuracy, thus making mesh generation

a complex and time consuming step in Finite Element analysis. For complicated

geometries, this worsens. In order to conquer this limitation, fictitious or embedded

domain methods have emerged. These are non-boundary conforming discretization

methods. One of these is the Finite Cell Method (FCM) introduced in [12]. As

mentioned in the introduction, this is an immersed boundary method based on higher

19


order finite elements wherein the physical domain Ωphy is embedded into a larger

embedding domain or fictitious domain Ωfict, such that the union Ω∪ of these yeilds

a simple geometry that can easily be meshed using a Cartesian grid as depicted in

Fig:2.7. The actual geometry is retrieved during the integration processes of the weak

form. If the integration is performed with sufficient accuracy, the FCM inherits the

convergence properties of the p-version. This section is dedicated to explain briefly

the basic concepts of the FCM.

Figure 2.7: Basic concept of the Finite Cell Method [19]

2.4.1 Basic concept

In Finite Cell Method, the process of mesh generation and solution approximation

are separated. This enables independent discretization of geometry and solution field.

Thus in FCM, there is a solution mesh which is used to approximate the solution

which does not resolve the geometry of the domain and an integration mesh which

is independent of the solution mesh and is used to resolve the geometry. Since the

integration mesh is used to represent the geometry and has nothing to do with the

solution, it does not introduce additional degrees of freedom.

Firstly, to recover the original boundary boundary value problem on Ω∪, on indicator

function α(x ) is defined as,

α(x ) =

1 : ∀x ∈ Ωphy

ε : ∀x /∈ Ωphy(2.13)

where ε is a small value used to ensure numerical stability.

The weak form is multiplied by this indicator function thus eliminating the contribu-

tions from the fictitious domain. This procedure for the phase-field problem under

consideration is formulated in the next chapter under section 3.4

20


In FCM, the difficulty of geometry resolution is shifted from discretization level to

integration level. Therefore, in the second step, the original domain geometry is

recovered using a separate integration mesh. Due to the discontinuity introduced by

the indicator function α(x ) that is used to impose a penalty on the fictitious domain,

a standard Gauß-Legendre quadrature is not the optimal numerical integration scheme.

A suitable integration scheme is the space-tree approach where recursive sub-division

of cut cells is carried out to accurately integrate the discontinuous integrals. In 2D,

this space-tree subdivision method is called quadtree and octree in 3D. The results

presented in this thesis make use of quadtree integration scheme [19] as shown in

Fig:2.8. In this figure, k referes to the partitioning depth. Though quadtree approach is

robust, it generates a large number of quadrature points and hence more sophisticated

schemes have been developed for better efficiency: smart-octree [13], moment-fitting

[14], quadratic re-parameterization for terahedral finite cell method [15] and adaptively

weighted quadratures [16].

Figure 2.8: Quadtree integration [19]

2.4.2 Application of non-boundary conforming boundary

conditions

Since the Finite Cell Method modifies the actual physical domain to get a different

computational domain, boundaries of the physical domain no longer conform with

the Finite Element mesh. This makes application of boundary conditions challenging.

Application of the Neumann boundary condition involve evaluating a boundary integral

on the right hand side of the weak form. But since the boundary is not resolved by

21


the mesh in FCM, Neumann boundary conditions are applied by an explicit surface

discretization.

On the other hand, enforcing the Dirichlet boundary conditions is involved. Mere

manipulation of entries related to Dirichlet boundary conditions in the stiffness matrix

cannot be done due to unresolved boundary, instead they are imposed weakly. To

meet this end, there are different ways in which Dirichlet boundary conditions can be

imposed, for example, the penalty method or Nitsche’s method [18]. These methods

extend the weak formulation to include the Dirichlet boundary conditions in a weak

sense. In this work, the penalty approach is followed and its formulation is presented

in section 3.4. Similar to the application of Neumann boundary conditions, the

constraining expressions are evaluated using a separate surface integration mesh which

does not introduce any additional degrees of freedom.

22

3 Formulation

Consider an arbitrary shaped domain Ω ∈ Rd with a boundary ∂Ω. Suppose that this

body has a crack boundary denoted by Γ as shown in Fig.3.1. Let u(x ,t) ∈ Rd be the

displacement of a point x ∈ Ω at time t. The displacement field satisfies the Dirichlet

boundary condition ui(x ,t) = gi(x ,t) on ∂Ωgi and Neuman boundary condition σijnj= hi on ∂Ωhi

. Note that i, j = 1,2,....d are the indices for components of vectors and

tensors represented in Einstein summation notation.

Γ

Ω

∂Ωgi

∂Ωhi

Figure 3.1: Schematic representation of the domain Ω with internal discontinuity Γ .

The following definition for strain tensor in the case of small deformation is used in

this work.

εij = ui,j =1

2

(∂ui∂xj

+∂uj∂xi

)(3.1)

The total potential energy of the body is the sum of elastic energy and fracture energy

and is given by

Ψpot(u ,Γ) =

∫Ω

Ψe(ε(u))dx +

∫Γ

Gcds (3.2)

where Ψe is the elastic energy density function and Gc is the fracture toughness as

introduced in Eq:(2.2).

The kinetic energy of the body is defined as

Ψkin(u) =1

2

∫Ω

ρuiuidx (3.3)

where u = ∂u∂t , ρ is the mass density of the material of the body.

23

3 Formulation

3.1 Quasi-static brittle fracture

As discussed in section 2.2, a phase-field parameter s is introduced to approximate

the discontinuous crack surface smoothly. Following the phase-field brittle fracture

model based on variational formulation introduced by Francfort and Marigo [22], the

quasi-static crack initiation and propagation process is governed by the minimization

of the free energy functional given by,

E(u ,Γ) = Ψpot(u ,Γ) =

∫Ω

Ψe(ε(u))dx +

∫Γ

Gcds (3.4)

For efficient numerical implementation, the energy functional is regularized by Bourdin

et al [21],

E(u , s) ≈∫

Ω

[Ψe(ε, s) +Gc

(1

4l(1− s)2 + l|∇,xs|2

)]dx (3.5)

where, the elastic strain energy density is degraded by a phase-field based penalty as

in Eq:(3.6) and the fracture energy is approximated as in Eq:(3.7).

Ψe(ε) ≈ Ψe(ε, s) = g(s)Ψe(ε) (3.6)

∫Γ

Gcds ≈∫

Ω

Gc

[1

4l(1− s)2 + l|∇,xs|2

]dx (3.7)

In Eq:(3.6), g(s) is called the stress degradation function. It is necessary for the stress

degradation function to satisfy the following conditions [24] :

g(0) = 0

g(1) = 1

g(s) > 0 for s 6= 0

g′(0) = 0

g′(1) > 0

As mentioned earlier, s is 0 at points which are fully damaged and 1 at which the

material is undamaged, see Eq:(2.9). To satisfy this, it is necessary to fulfill the first

two conditions. The third condition ensures that the phase field parameter goes to

zero if and only if the material point is fully damaged. The fourth condition is to

24

3 Formulation

be satisfied to keep intact the principle of conservation of energy. Lastly, the fifth

condition imposes that all the material points in the domain are undamaged at an

arbitrary initial state. It is important to satisfy the above conditions in order to

achieve Γ convergence as explained by Braides [25] and to ensure reasonable evolution

of stress and crack field. From literature, g(s) = (1 − η)s2 + η , which satisfies all

the requirements, has been used in this work (η is a dimensionless parameter used to

model an artificial stiffness of a completely damaged phase) [23].

Two ways of degrading the elastic strain energy density is followed in the field of

phase-field modeling. In isotropic formulation, the entire elastic strain energy density

is multiplied by the stress degradation function g(s) as mentioned in Eq:(3.6). The

second type of formulation is the one which has a tensile-compression split of strain

energy density and only the tensile part is degraded. In this work, one of the variants

with a tensile-compression split presented by Miehe et al [7] is followed.

Anisotropic Miehe formulation

The anisotropic Miehe formulation presented by Miehe et al [7] is considering a tensile

and compression split is anisotropic by nature which gives the formulation its name.

In this formulation, the free energy funtional defined in Eq:(3.5) is modified as,

E(u , s) =

∫Ω

[g(s)Ψ+

e (ε) + Ψ−e (ε) +Gc

(1

4l(1− s)2 + l|∇,xs|2

)]dx (3.8)

where, the strain energy density is approximated as,

Ψe(ε) ≈ Ψe(ε, s) = g(s)Ψ+e (ε) + Ψ−e (ε) (3.9)

Ψ+e and Ψ−e are the tensile and compressive components of elastic strain energy density,

respectively, obtained by the spectral decomposition of the strain tensor and are defined

as,

Ψe+ =

1

2λ〈tr(ε)〉2 + µtr[(ε+)2] (3.10)

and

Ψ−e =1

2λ[(tr(ε)− 〈tr(ε)〉)2

]+ µtr[(ε− ε+)

2] (3.11)

where λ and µ are Lame constants and

〈χ〉 =

χ : χ > 0

0 : χ ≤ 0(3.12)

25

3 Formulation

Let the strain tensor be defined as,

ε = PAPT (3.13)

where P consists of the orthonormal eigenvectors of ε and A is a diagonal matrix

of principal strains λi. The following definitions for tensile and compressive split of

strains are used for spectral decomposition.

ε+ = PA+PT (3.14)

ε− = PA−PT (3.15)

where, A+ = diag(〈λi〉) is a diagonal matrix and A− = A−A+.

This split in strain energy density allows degradation of tensile component of strain

energy density only, thereby inhibiting crack growth under compression. Selective

degradation of strain energy density is important for dynamic simulation in particular

since the stress waves reflecting from domain boundaries may create unphysical fracture

patterns otherwise, refer [23] for details.

The Euler-Lagrange equations of the minimization of the energy functional Eq:(3.8)

leads to the strong form where the governing equation for phase-field is driven by the

elastic strain energy density.

strong form

∂σij

∂xj+ ρb = 0[

4lGc

(1− η)H + 1

]s − 4 l2 ∆s = 1 on Ω

(3.16)

where, b is the body force and σij = g(s)∂Ψ+e

∂εij+ ∂Ψ−

e

∂εijand

Ht∈[0,τ ]

= max Ψ+e (ε, t) (3.17)

is called the history variable. It represents the maximum tensile elastic strain energy

density. This treatment enables decoupling of the two differential equations and hence

a staggered scheme can be used. Also, the irreversibility condition Γ(t) ⊆ Γ(t + dt)

at any time (psuedo-time in the case of quasi-static fracture) t is enforced using this

history variable H.

To sum up, the strong form for quasi-static brittle fracture problem is represented by

two partial differential equations Eq:(3.16) which are solved for u(x ) and s(x ) using

26

3 Formulation

staggered scheme as explained earlier. They are subjected to the following boundary

conditions,

boundary conditions

ui = gi on ∂Ωgi

σijnj = hi on ∂Ωhi

∂s∂xini = 0 on ∂Ω

(3.18)

Hybrid fomulation

The isotropic formulation where the entire elastic strain energy density is degraded

by the phase-field although numerically inexpensive due to its linear nature, leads to

unrealistic crack patterns since it allows crack propagation under compression. This is

overcome by anisotropic formulations by introducing a split in elastic strain energy

density which differentiates between tension and compression. However, this split is

anisotropic in nature. This means that within each staggered step, the momentum

equation is non-linear which is to be solved by incrementation techniques and hence is

numerically expensive. In order to take an advantage of the linear nature of the isotropic

formulation whilst preserving the tension-compression differentiation, Ambati et al [5]

proposed the so-called hybrid formulation. In this formulation, the tension-compression

split is retained for the calculation of strain energy density and thus Eq:(3.17) remains

the same. However, the decomposition that exists in stress calculation is let go,

σij = g(s)∂Ψe

∂εij(3.19)

This model hence retains the linear nature of the momentum equation and reduces

the computational cost significantly.

3.2 Extension to dynamic brittle fracture

The additional term to be considered for dynamic fracture is the kinetic energy

represented by Eq:(3.3). The Lagrangian for the fracture problem, Eq:(3.20), is

formed by combining the potential and kinetic energies incorporating the phase-field

approximations explained earlier. This Lagrangian based Euler-Lagrange equations

govern the motion of the body and the evolution of crack which are presented in this

27

3 Formulation

section and are adapted to phase-field setting. The fracture energy approximation

remains the same as shown in Eq:(3.7)

Lε(u , u , s) =

∫Ω

[12ρuiui− g(s)Ψ+

e (ε)−Ψ−e (ε)]dx −

∫Ω

Gc[ 1

4l(1− s)2 + l|∇,xs|2

]dx

(3.20)

The Euler-Lagrange equations of the minimization of the Lagrangian represented by

Eq:(3.20) leads to the strong form. Similar to the quasi-static case, the strong form for

dynamic brittle fracture problem is represented by two partial differential equations

which are solved for u(x , t) and s(x , t) using staggered scheme as explained earlier.

strong form

∂σij

∂xj+ ρb = ρui on Ω[

4lGc

(1− η)H + 1

]s − 4 l2 ∆s = 1 on Ω

(3.21)

These equations are subjected to the following boundary conditions and initial condi-

tions.

boundary conditions

ui = gi on ∂Ωgi

σijnj = hi on ∂Ωhi

∂s∂xini = 0 on ∂Ω

(3.22)

intial conditions

u(x , 0) = u0(x ) x ∈ Ω

u(x , 0) = v0(x ) x ∈ Ω(3.23)

3.3 Numerical formulation

In order to numerically solve the initial boundary value problem of the brittle fracture

process, it is necessary to derive a weak form and then discretize the same. This section

is dedicated to the weak formulation of the governing equations for both quasi-static

and dynamic cases, first in a continuous domain and then in Galerkin setting in order

to facilitate finite element implementation.

28

3 Formulation

3.3.1 Quasi-static brittle fracture

Continuous problem in the weak form

In the following, the necessary spaces for the derivation of the weak form are defined.

Suppose S be the trial space for displacement solution and S be the trial space for

phase-field solution. Similarly V and V are the spaces for the test functions.

S = u(x ) ∈ H1(Ω)| ui = gi on ∂Ωgi (3.24)

S = s(x ) ∈ H1(Ω) (3.25)

V = w(x ) ∈ H1(Ω)| wi = 0 on ∂Ωgi (3.26)

V = q ∈ H1(Ω) (3.27)

where H1 is a Hilbert space.

The weak form is obtained by multiplying the strong form by appropriate test functions

and then applying Green’s theorem. It states, given g(x ), h(x ), b(x ) , find u(x ) ∈S and s(x ) ∈ S such that,

weak form in continuous domain

(σ,∇w)Ω = (ρb,w)Ω + (h ,w)∂Ωh

([

4lGc

(1− η)H + 1]s, q)

Ω+ (4l2∇s,∇q)Ω = (1, q)Ω

(3.28)

where (., .)Ω is the L2 inner product on Ω. The above weak form holds ∀w ∈ V , q ∈ V

Galerkin form

The previously introduced spaces to which the solution and test functions belong are

infinite. For FE analysis, as the name suggests, it is necessary to logically reduce these

spaces to valid finite ones. Galerkin method is followed for this purpose. Let Sh ⊂ S,

Sh ⊂ S, Vh ⊂ V and Vh ⊂ V be the finite dimensional approximating spaces. The

Galerkin form states, find uh(x ) ∈ Sh and sh(x ) ∈ Sh such that,

Galerkin form

(σ,∇wh)Ωh = (ρbh,wh)Ωh + (hh,wh)∂Ωh

h

([

4lGc

(1− η)H + 1]sh, qh)

Ωh+ (4l2∇sh,∇qh)Ωh = (1, qh)Ωh

(3.29)

In finite element method, the continuous domain Ω is divided into finite number of

elements. This is denoted by Ωh. The discretized solutions, displacement uh(x ) and

phase field sh(x ) and the trial functions, wh(x ) for displacement field and qh(x ) for

29

3 Formulation

phase field are assumed to be linear combination of basis functions. Here, Bubnov-

Galerkin formulation is followed, where same basis functions are used for both solutions

and the trial functions.

u ≈ uhi =

nb∑A

NA(x )diA (3.30)

w ≈ whi =

nb∑A

NA(x )ciA (3.31)

s ≈ sh =

nb∑A

NA(x )φA (3.32)

q ≈ qh =

nb∑A

NA(x )χA (3.33)

where nb is the dimension of the discrete space Ωh, NA are the basis functions, i is the

spatial degree of freedom (nodes), diA, ciA, φA, χA are the nodal values.

Subtituting these in the Galerkin form in Eq:(3.29), the following system is obtained.

Kuu∆d = Fu (3.34)

Kuu = KAB,i (3.35)

KAB,i = (σjk, BijkA )

Ωh (3.36)

Fu = FA,i (3.37)

FA,i = (ρbh, NAe i)Ωh + (hh, NAe i)∂Ωhh

(3.38)

where e i is the i th Euclidean basis vector and

BijkA =

1

2

(∂NA∂x j

δik +∂NA∂x k

δij

)σjk = g(s)

∂Ψ+e

∂εjk+∂Ψ−e∂εjk

for the anisotropic Miehe formulation

σjk = g(s)∂Ψe

∂εjkfor the hybrid formulation

Similarly, the phase-field arrays are defined.

30

3 Formulation

Kss∆φ = Fs

Kss = KAB

KAB = ([ 4l

Gc(1− η)H + 1

]NB, NA)

Ωh

+ (4l2∂NB∂x i

,∂NA∂x i

)Ωh

Fs = FAFA = (1, NA)Ωh

In the case of the anisotropic Miehe formulation, due to the tension-compression

split of stresses, the governing equation system for the displacement field, Eq:(3.34),

is non-linear. Generally, such equations are solved using incrementational solution

techniques which required a consistent linearization of all the terms in the weak form.

In this work, followed is the Newton-Raphson method, see Ambati [26] for detailed

formulation.

3.3.2 Dynamic brittle fracture

The weak form for dynamic brittle fracture states, given g , h , u0, u0 and s0, find

u(x , t) ∈ S and s(x , t) ∈ S such that,

weak form in continuous domain

(ρu,w)Ω + (σ,∇w)Ω = (ρb,w)Ω + (h ,w)∂Ωh

([

4lGc

(1− η)H + 1]s, q)

Ω+ (4l2∇s,∇q)Ω = (1, q)Ω

(ρu(0),w)Ω = (ρu0,w)Ω

(ρu(0),w)Ω = (ρu0,w)Ω

(s(0), q)Ω = (s0, q)Ω

(3.39)

The semi-discrete Galerking form for numerical implementation states, find uh(x , t) ∈Sht and sh(x , t) ∈ Sht such that,

semidiscrete Galerkin form

(ρuh,wh)Ωh + (σ,∇wh)Ωh = (ρbh,wh)Ωh + (hh,wh)∂Ωh

h

([

4lGc

(1− η)H + 1]sh, qh)

Ωh+ (4l2∇sh,∇qh)Ωh = (1, qh)Ωh

(ρuh(0),wh)Ωh = (ρu0,wh)Ωh

(ρuh(0),wh)Ωh = (ρu0,wh)Ωh

(sh(0), qh)Ωh = (s0, qh)Ωh

(3.40)

The above is called semi-discrete since the discretization is only in space and not

in time. When compared to the Galerkin form of the quasi-static formulation, the

semi-discrete form is similar but with an additional mass matrix.

31

3 Formulation

Time discretization

As mentioned earlier, staggered time integration scheme is followed to solve the dynamic

problem. Recall that in staggered scheme, momentum and phase-field equations are

solved independently. The momentum equation is solved at a given time step, the

displacements in the phase-field equation is updated and then the equation is solved.

Assuming negligible body forces. the following definition of residuals are necessary for

this scheme.

Ru = RuA,i (3.41)

RuA,i = (h , NAe i)∂Ωhh− (ρuh, NAe i)Ωh − (σjk, B

ijkA )

Ωh (3.42)

Rs = RsA (3.43)

RsA = (1, NA)Ωh − ([ 4l

Gc(1− η)H + 1

]sh, NA)

Ωh

− (4l2∂sh

∂x i,∂NA∂x i

)Ωh

(3.44)

After linearization of the momentum equation, predictor-corrector method is then

applied to solve this system, see [23] for details.

3.4 Finite Cell Method for phase-field quasi-static

brittle fracture

As introduced in section 2.4, the Finite Cell Method is an embedded domain method.

The physical domain is recovered using the indicator function α as defined in Eq:(2.13).

In the following, presented is the weak formulation for phase-field quasi-static brittle

fracture modified for the Finite Cell Method implementation.

Momentum Equation

The momentum equation for quasi-static brittle fracture, Eq:(3.28), is modified for

the Finite Cell Method implementation and is as follows. Penalty method is used to

impose the Dirichlet boundary condition with β being the penalty parameter, typically

of the order 1010 to 1012.

(σ,∇w)Ωphy+ (εσ,∇w)Ωfict

+ (βu ,w)∂Ωg

= (ρb,w)Ωphy+ (h ,w)∂Ωh

+ (βg ,w)∂Ωg

(3.45)

(ε 4l2∇s,∇q)Ωfict

= (1, q)Ωphy(3.46)

32

3 Formulation

Phase-field Equation

Similar to the momentum equation, the weak form of the phase-field equation modified

for FCM is as presented below. Due to homogeneous Neumann boundary condition

throughout the boundary, enforcing the boundary conditions for this initial boundary

value problem is trivial.

([ 4l

Gc(1− η)H + 1

]s, q)

Ωphy

+ (ε[ 4l

Gc(1− η)H + 1

]s, q)

Ωfict

+ (4l2∇s,∇q)Ωphy+ (ε 4l2∇s,∇q)Ωfict

= (1, q)Ωphy

(3.47)

(Note that the Finite Cell formulation for dynamic brittle frature follows the same

procedure resulting in similar form with an additional mass matrix term related to the

kinetic energy.)

33

4 Numerical results

In this chapter, the numerical performance of phase-field fracture model combined

with the uniform multi-level hp-refinement and the Finite Cell Method is investigated.

Illustrative examples for quasi-static and dynamic brittle fracture are presented. Inte-

grated Legendre polynomials are used as basis functions for discretization and p+1

integration rule is adopted for numerical integration. In each example, history variable

is stored at geometric points using voxel domain geometry modeling technique and the

same is used for the definition of pre-existing crack. Two separate discretizations are

defined, one for the elastic problem and one for the phase-field problem. Two different

refinement strategies are followed for adaptive refinement of the two problems. In

every load or time step, the base mesh for the elastic problem is refined in the vicinity

of the crack. It is refined in those areas of the physical domain where the phase-field

values reach a certain threshold limit. The phase-field problem is then refined such

that it follows the discretization of the elastic problem.

The chapter starts off with two quasi-static benchmark problems, single-edge notched

tension test and single-edge notched shear test. The current implementation of the

hybrid formulation is validated by qualitative and quantitative comparison of the

obtained results for single-edge notched tension test to the results obtained by Ambati

et al [5] under similar settings. The anisotropic Miehe formulation is validated using

single-edge notched shear test. Further, for these two examples, convergence study for

both the anisotropic Miehe and the hybrid formulations are presented.

This is followed by a more complex example, a notched plate with hole, which demon-

strates the idea of combining Finite Cell Method with the concept of phase-field

modeling. A non-geometry conforming mesh is used in this case. This example is used

to study the effect of number of staggered iterations used within every displacement step.

As a concluding example for the quasi-static brittle fracture case, a non-pre-cracked

cocnrete block subjected to compressive load is studied.

The next section deals with a dynamic crack branching problem under uniaxial

tension. Numerical results obtained using both the anisotropic Miehe and the hybrid

formulations are discussed.

4.1 Single-edge notched tension test

Consider a two-dimensional sqaure plate of side 1mm with a pre-existing horizontal

crack at its mid height as shown in Fig:4.1. The pre-existing crack is modeled using the

history variable as in Eq:(4.1). This notched plate is subjected to a constant vertical

34

4 Numerical results

displacement on its top edge. The material and model parameters used are: E = 210

GPa, ν = 0.3, (λ = 121.15 kN/mm2 and µ = 80.77 kN/mm2) Gc = 0.0027 kN/mm, l

= 0.004mm, η = 10−6.

H(x ) = B

Gc

4l : 0 ≤ x1 ≤ 0.5, (0.5− l) ≤ x2 ≤ (0.5 + l)

0 : otherwise(4.1)

0.5mm

0.5mm

0.5mm 0.5mm

u

Figure 4.1: Geometry and boundary conditions for single-edge notched tension test

4.1.1 Validation of implementation of the hybrid formulation

To validate the current implementation of the hybrid formulation, consider a dis-

cretization with 4 × 4 elements, ansatz order p = 4 and a refinemenet depth k = 5.

Displacement control with a displacement increment of ∆u = 1 × 10−6mm and two

staggered iteration for every load step are used in oder to match the parameters of

the same example in Ambati et al [5]. The number of DOFs at the beginning of the

simulation is 12906 and 33258 at a displacement value of 7.0×10−3mm. The reference

result uses a fixed mesh with bilinear shape functions which is apriori refined in the

location where crack is known to develop and the crack has been defined using the

geometry instead of the history variable. The phase-field evolution representing crack

propagation is shown in Fig.4.2 and the corresponding load-displacement curve in

Fig.4.3. It can be observed that the crack patterns and the load-displacement curves

almost coincide, thus validating the implementation.

35

4 Numerical results

Ref

eren

cep

=4,k

=5

(a) u = 5.5×10−3mm (b) u = 5.7×10−3mm (c) u = 6.0×10−3mm

Figure 4.2: Single-edge notched tension test. Validation of crack evolution with hybrid

formulation against reference results.

0 1 2 3 4 5 6

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

reference solution

p = 4, k = 5

Figure 4.3: Single-edge notched tension test. Validation of load-displacement behavior

with hybrid formulation against reference results.

36

4 Numerical results

4.1.2 Parametric influence of convergence behavior of the

phase-field problem

In the following, convergence behavior of the phase-field solution is studied under two

scenarios. To this end, firstly, three discretizations with linear, quadratic and cubic

ansatz functions and a grid of 4 × 4 base elements are chosen. The material and model

paramters remain the same except that a larger displacement increment of ∆u = 1 ×10−5mm is used in order to shorten the run time. The only disadvantage of using larger

displacement increments is that the sharp drop in the reaction forces at peak values,

i.e,. the catastrophic crack propagation phenomenon is not well captured. Fig.4.4

depicts snapshots of crack path at different displacements. A common observation in

the results of both the formulations is that for linear shape functions, a much higher

displacement value leads to crack propagation compared to that using higher order

shape functions. The same trend is reflected in the load-displacement plots in Fig.4.5

where the peak force value using linear basis is higher and is shifted towards a higher

displacement value. In Fig.4.5, a converging behavior is observed as the order of the

polynomial is increased, where error between the results using quadratic and cubic

basis is much lower than that between linear and quadratic basis.

p=

1p

=2

p=

3


(a) u = 6.5×10−3mm (b) u = 7.0×10−3mm (c) u = 7.3×10−3mm

37

4 Numerical results

p=

1p

=2

p=

3

Hybrid formulation

(a) u = 6.5×10−3mm (b) u = 7.0×10−3mm (c) u = 7.3×10−3mm

Figure 4.4: Single-edge notched tension test. Multi-level hp-refinement for different

ansatz orders with k = 6. Crack phase-field at different displacements.

0 1 2 3 4 5 6 7 8

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p = 1

p = 2

p = 3

reference solution

(a) Anisotropic Miehe formulation

38

4 Numerical results

0 1 2 3 4 5 6 7 8

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

p = 1

p = 2

p = 3

reference solution

(b) Hybrid formulation


ansatz orders with k = 6. Load-displacement curves.

To investigate the minimum required refinement depth for a given ansatz order, consider

a grid with 4 × 4 elements, ansatz order p = 3 and three different depths 4, 5 and

6. The results obtained for both the aniostropic Miehe and the hybrid formulation

are presented. Fig.4.6 depicts the phase-field crack evolution and Fig.4.7 shows the

corresponding load-displacement curves. At the outset, even though the crack path

is as expected for all cases, one can observe a significant difference in the phase-field

evolution for the lower refinement case. For a refinement depth of 4, crack propagation

is rather slow in the results of both the formulations. This is also reflected in the

load-displacement behavior where the load values with k = 4 show an overestimated

behavior. For both the formulations, excellent convergence can be observed as the

refinement depth is increased. It is worth noting that the solutions obtained using

multi-level hp-refinement for hybrid formulation almost coincide for the corresponding

results using the anisotropic Miehe formulation. Thus, it is clear from the above

two studies that, with different ansatz orders and with different refinement depths,

it is important to resolve the mesh carefully in order to avoid mesh related effects.

To conclude, it seems from the above parametric study that a dynamically adaptive

discretization with 4 × 4 elements, cubic shape functions and a refinement depth of 5

with around 7000 initial DOFs leading to around 18500 DOFs at the end of fracture

with run time of 50 minutes is sufficient to capture the fracture process. To obtain

similar quality results, a non-dynamically adaptive grid with 26058 DOFs has been

used in the reference results.

39

4 Numerical results

k=

4k

=5

k=

6


k=

4k

=5

k=

6

Hybrid formulation

(a) u = 6.5×10−3mm (b) u = 6.75×10−3mm (c) u = 7.0×10−3mm


refinement depths with p = 3. Crack phase-field evolution at different

displacements. 40

4 Numerical results

0 1 2 3 4 5 6 7 8

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

k = 4

k = 5

k = 6

reference solution


0 1 2 3 4 5 6 7 8

10-3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

k = 4

k = 5

k = 6

reference solution



refinement depths with p = 3. Load-displacement curves.

41

4 Numerical results

4.2 Single-edge notched shear test

0.5mm

0.5mm

0.5mm 0.5mm

u

Figure 4.8: Geometry and boundary conditions of single-edge notched shear test

4.2.1 Validation of implementation of the anisotropic Miehe

formulation

The same square plate with a pre-existing crack used for tension test is now subjected

to shear load as shown in Fig:4.8. The material and model parameters remain the

same as that for the tension test example. Firstly, implementation of the anisotropic

Miehe formulation is validated using this example. A displacement increment ∆u = 1

× 10−5mm is applied on a grid of 4 × 4 elements, ansatz order p = 4 and a refinement

depth k = 5 are used for this purpose. While the reference solution obtained by Ambati

et al [5] is using a relatively fine uniform mesh, the result obtained here is using an

adaptively refined mesh which changes over the course of the simulation from an intial

DOFs of 12906 to final DOFs of 41754. The crack patterns at different displacements

and the load-displacement curves are shown in Fig.4.9 and Fig.4.10, respectively. The

crack propagation path obtained using multi-level hp dynamically adaptive refinement

mostly coincides with that from the reference. However, a faster crack propagation

can be observed in this adaptive result in comparison to the reference results. This

relatively small difference, to the knowledge of the author, could be due to mesh related

effects. The reference solution is also a numerical one in which mesh related effects

cannot be completely eliminated.

42

4 Numerical results

Ref

eren

cep

=4,k

=5

(a) u = 0.012mm (b) u = 0.015mm (c) u = 0.020mm

Figure 4.9: Single-edge notched shear test. Validation of crack evolution with the

anisotropic Miehe formulation against reference results.

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

reference solution

p = 4, k = 5

Figure 4.10: Single-edge notched shear test. Validation of load-displacement behavior

with the anisotropic Miehe formulation against reference results.

43

4 Numerical results

4.2.2 Parametric influence of convergence behavior of the

phase-field problem

In this section, convergence study is carried out for different discretizations, similar to

the previous example. For this, a larger displacement increment of ∆u = 1 × 10−4mm

is used in order to shorten the computational time. Although this has no significant

effect on pre-peak part of the load-displacement curve, the post-peak curve varies. The

immediate drop of load values post-peak is not observed in this case similar to the

observation in the previous example.

A study of influence of ansatz order for a given refinement depth is carried out here.

Three case each, for both the anisotropic and the hybrid formulation, with ansatz

orders 1, 2 and 3, all with a refinement depth k = 6 and a grid with 4 × 4 elements are

studied. Result obtained by Ambati et al [5] for the same displacement increment value

is used as a reference. The crack pattern and the load-displacement curves for these

cases are depicted respectively in Fig.4.11 and Fig:4.12. The crack propagation path

seem to be almost the same for different ansatz orders. However, a careful observation

reveals small disturbances in the crack path with lower order polynomials. Similar to

the observation made in the tension test examples, the reaction force values are over

estimated at lower order ansatz functions. But excellent convergence is observed as

the order of the basis is increased.

p=

1p

=2

p=

3


(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm

44

4 Numerical results

p=

1p

=2

p=

3

Hybrid formulation

(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm

Figure 4.11: Single-edge notched shear test. Multi-level hp-refinement for different

ansatz orders with k = 6. Crack phase-field at different displacements.

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

p = 1

p = 2

p = 3

reference solution


45

4 Numerical results

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

p = 1

p = 2

p = 3

reference solution



ansatz orders with k = 6. Load-displacement curves.

To study the influence of refinement depth for a given ansatz order, consider a

discretization with 4 × 4 elements, ansatz order p = 5 and three different depths 3,

4 and 5. Both the aniostropic Miehe and the hybrid formulation are considered for

this study. Fig.4.13 depicts the phase-field crack evolution and Fig.4.14 shows the

corresponding load-displacement curves. Although the crack propagation path is well

defined and close to the expected one for the hybrid formulation, the displacement

required for the same amount of crack growth varies with refinement depth. This is

due to insufficient refinement which makes the solution badly resolved. The effect of

discretization on the solution is pronounced in the case of anisotropic Miehe formulation

where inadequate resolution leads to incorrect results. A crack can be seen propagating

at the boundaries which is unphysical.

Inference from the above two study cases is that for cubic shape functions, a refinement

depth of 6 with 21032 initial and 71834 final DOFs, approximate run time of 2 hours,

10 minutes or for a 5th order polynomial ansatz, a refinement depth of 5 with 20312

initial and 69032 final DOFs, with an approximate run time of 1 hour, 55 minutes,

eliminate major numerical artifacts.

46

4 Numerical results

k=

3k

=4

k=

5


k=

3k

=4

k=

5

Hybrid formulation

(a) u = 0.012mm (b) u = 0.02mm (c) u = 0.025mm


refinement depths with p = 5. Crack phase-field at different displacements.

47

4 Numerical results

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

k = 3

k = 4

k = 5

reference solution


0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

k = 3

k = 4

k = 5

reference solution



refinement depths with p = 5. Load-displacement curves.

4.2.3 Comparative study between uniform and adaptively refined

meshes

The objective of this section is to study the performance of a globally refined and locally

refined discretizations for the same order of the shape functions. For this, consider

the shear test specimen with the same material parameters as before. A displacement

increment of ∆u = 1 × 10−4mm is used. In order to resolve a length scale parameter

of l = 0.004 mm using a globally refined mesh, a grid with a minimum of 250 × 250

48

4 Numerical results

elements are required, resulting in 126002 degrees of freedom. The resources required

to run such cases are very high and time consuming. Hence, a larger length scale

parameter is chosen which results in a thicker crack due to the larger transition zone

between damaged and undamaged regions.

Firstly, the specimen is uniformly discretized into a grid of 128 × 128 elements with

linear ansatz such that the smallest element is 0.0078 mm × 0.0078 mm is size. This

descretization is sufficient to resolve a length scale parameter of l = 0.01 mm. In order

to compare the results obtained from uniform mesh to that from an adaptively refined

mesh using uniform multi-level hp-refinement, consider two cases with 32 × 32 with

2 multi-level hp-refinements and 16 × 16 with 3 multi-level hp-refinements. These

two discretizations ensure that the smallest element size remains the same as that

of the uniform grid. Fig.4.15 shows the crack evolution over displacements and the

corresponding load-displacement curve is shown in Fig.4.16. They clearly demonstrate

that the same quality of the solution can be achieved by using coarser base mesh if the

refinement depth is increased appropriately. In Table 4.1, the number of degrees of

freedom and the run time comparison for the three cases under consideration is done.

The result values reflect the increasing efficiency in terms of number of unknows and

the run time when the refinement is made more local at regions with crack while using

larger elements where away from the crack where the solution is mostly smooth.

128×

128

elem

s32×

32el

ems

16×

16el

ems

(a) u = 0.012mm (b) u = 0.015mm (c) u = 0.020mm

Figure 4.15: Single-edge notched shear test. Comparison of crack phase-field for uni-

formly and adaptively refined meshes with p = 1 at different displacements.49

4 Numerical results

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

128 X 128 elements

32 X 32 elements, k = 2

16 X 16 elements, k = 3

Figure 4.16: Single-edge notched shear test. Comparison of load-displacement curves

for uniformly and adaptively refined meshes with p = 1.

Case Initial DOFs Final DOFs Run time

128×128 elements 33828 33828 2 hours, 45 minutes

32×32 elements, 2 multi-level hp-ref 2678 6326 30 minutes

16×16 elements, 3 multi-level hp-ref 1168 4884 16 minutes

Table 4.1: Single-edge notched shear test. Parametric comparison for uniformly and

adaptively refined meshes with p = 1 and the hybrid formulation.

50

4 Numerical results

4.3 Notched plate with hole

0.55mm

0.65mm

uy = 0

0.2mm

0.2mm

0.51mm

1.2mm

0.365mm

0.2mm

0.2mm

0.1mm

x

y

0.2mm

0.1mm

0.1mm

ux = 0

ux = 0

uy = u

Boundary Conditions

Figure 4.17: Geometry and boundary conditions of notched plate with hole

Figure 4.18: Notched plate with hole. Crack pattern from the experiment

This example demonstrates the application of the FCM in the context of phase-field

models. Here, a notched plate with the geometry and boundary conditions as shown

in Fig.4.17 is simulated. The material and model properties used for this example are:

51

4 Numerical results

E = 6 GPa, ν = 0.22, (λ = 121.15 kN/mm2 and µ = 80.77 kN/mm2) Gc = 0.00227

kN/mm, l = 0.005mm, η = 10−6 and a displacement increment of ∆u = 5 × 10−4mm

is applied upto 0.1mm. A discretization with 15 × 30 elements, quadratic integrated

Legendre polynomials and refinement depth of 4 is used. The indicator function value

is set to 10−8 in the fictitious domain, the three holes in this example. A partitioning

depth of 3 is made use in numerical integration using quadtree integration scheme.

Crack patterns obtained for both the anisotropic Miehe and the hybrid formulations are

shown in Fig.4.19. It can be observed that the crack patterns are similar, except that

the anisotropic Miehe formulation predicts a larger crack growth at lower displacement

values when compared to the hrbrid formulation. Also, the crack pattern in the hybrid

formulation is inclined to the horizontal at a larger angle than that in the anisotropic

Miehe formulation. This inclined behaviour can be observed in the experimental results

obtained for a similar (scaled up in dimensions) specimen by Ambati et al [5], see

Fig.4.18.


Hybrid formulation

(a) u = 0.035mm (b) u = 0.05mm (c) u = 0.085mm (d) u = 0.1mm

Figure 4.19: Notched plate with hole. Multi-level hp-refinement: p = 2, k = 4. Crack

phase-field at different displacements.

52

4 Numerical results

4.3.1 Parametric influence of number of staggered iterations on

crack propagation

To study the influence of number of staggered iterations used per displacement incre-

ment step, consider the same example for three different staggered iteration value cases:

2 , 3 and 5. Hybrid formulation with cubic integrated Legendre shape functions are

used here, while rest of the parameters remain the same as explained above. Resulting

crack propagation is depicted in Fig.4.20. It can be observed that the number of

staggered iterations play an important role in crack propagation. This is because for

higher number of staggered iterations, the momentum equation and the phase-field

equation tend to attain equilibrium with each other. Hence, this necessitates the

definition of a proper stopping criterion which stops at a particular staggered iteration

that ensures equilibrium, refer [5] for details.

2st

agg

iter

3st

agg

iter

5st

agg

iter

(a) u = 0.035mm (b) u = 0.05mm (c) u = 0.085mm (d) u = 0.1mm

Figure 4.20: Notched plate with hole. Multi-level hp-refinement for different number

of staggered iterations : p = 3, k = 4. Crack phase-field at different

displacements.

53

4 Numerical results

4.4 Crack propagation under compressive loads

0.9m

0.35m

u0.045m

0.01m

Figure 4.21: Geometry and boundary conditions of concrete block under compression.

This example aims at studying the behavior of the phase-field model in the case of load

induced crack in a concrete block. Consider a crack-free concrete block with thickness

0.1m as shown in Fig.4.21. The material of the block has E = 38.992 GPa, ν = 0.18,

Gc = 148.5 N/m and is subjected to compression. This three dimensional scenario is

numerically solved as a two dimensional plane strain problem using phase-field model

with the following parameters: 5 staggered iterations, l = 0.01m, η = 10−6 and a

displacement increment of ∆ u = 5 × 10−5m. A discretization with linear 5 × 15

elements and 4 uniform multi-level hp-refinements are used. A pre-refinement is done

in the vicinity of the displacement boundary condition where the crack is expected to

initiate. Fig.4.22 and Fig.4.23 show the experimentally and numerically obtained crack

patterns, respectively. As evident, the numerical results are not similar to the ones

obtained from the experiment. Also, the results from the anisotropic and the hybrid

formulations do not comply with each other. With the hybrid formulation, no crack is

seem to be initiated. Larger displacements are observed to lead to unphysical results

and hence are not presented here. A few obersvations and remarks in this regard are

as follows:

In the anisotropic Miehe formulation, the number of Newton-Raphson iterations

taken for convergence is found to increase beyond 30 after a few load steps.

This worsens when body forces are added to the elastic initial boundary value

problem.

Despite using a large number of staggered iterations in order to ensure equilibrium

between the elastic and the phase-field problem, the convergence remains poor.

Not considering into account the body forces and the dynamic nature of the

problem could be one of the reasons for the propagation path of the crack to

54

4 Numerical results

deviate from the experimental one.

Further investigation needs to be done in order to study about applying the

phase-field model for such initial crack-free problems and is beyond the scope of

the thesis.

Figure 4.22: Concrete block under compression. Crack pattern from the experiments

carried out by Dipl.-Ing. Gerald Schmidt-Thro, Lehrstuhl fur Massivbau,

TU Munchen.


Hybrid formulation

(a) u = 0.5mm (b) u = 1mm (c) u = 1.5mm (d) u = 2.5mm

Figure 4.23: Concrete block under compression. Multi-level hp-refinement: p = 1, k =

4. Crack phase-field at different displacements.

55

4 Numerical results

4.5 Dynamic crack branching

Consider a rectangular specimen of length B = 0.1 m and H = 0.04 m with a pre-

existing horizontal crack at its mid height as shown in Fig:4.24. The specimen is

subjected to a constant uniaxial tension of σ(t). The material and model parameters

used for simulation are ρ = 2450 kg/m3, E = 32 GPa, Gc = 3 J/m2, ν = 0.2, η =

10−6. A discretization with 8 × 4 elements with two linear and quadratic ansatz, time

step of dt = 0.2µs and a length scale parameter of l = 2.5× 10−4m are used. Results

using two staggered iterations are presented in order to match the parameters used by

Borden et al [23] for the same benchmark example.

H

B

Initial crack, Γ

σ(t)

σ(t)

Boundary conditions toensure numerical stability

0.01m

Figure 4.24: Geometry and loading of dynamic uniaxial tension test.

Remark

When the problem is solved numerically, care should be taken while imposing

the boundary conditions in order to avoid artifacts. If rigid body motion of the

sample is not arrested, then the condition number of the stiffness matrix becomes

very high which might lead to a singular or a nearly singular matrix resulting in

unphysical behavior. In order to restrict the rigid body motion, in the example

presented here, the mid node on the right edge is fixed and a dirichlet boundary

condition is weakly imposed on a small segment on the mid-line as indicated in

Fig.4.24.

The predictor-corrector time integration scheme was found to convergence at a

very slow rate when the strong constrain at the mid node on the right edge was

imposed using a penalty.

56

4 Numerical results

The length of the segment along which the weak boundary condition is applied

has been empirically chosen to be 0.01m. Further optimization can be done in

this regard.

p=

1

k=

5

p=

2

k=

5


(a) t = 20µs (b) t = 50µs (c) t = 80µs

Figure 4.25: Dynamic crack branching under uniaxial tension. Multi-level hp-

refinement for different ansatz order with k = 5 using the anisotropic

Miehe formulation. Crack phase-field evolution over time.

0 10 20 30 40 50 60 70 800

0.05

0.1

0.15

0.2

0.25p = 1

p = 2

reference


refinement for different ansatz order with k = 5. Plot of strain energy

overtime.

57

4 Numerical results

The numerical results obtained using the anisotropic Miehe formulation are presented in

Fig.4.25 and Fig.4.26. Similar to the observations made in quasi-static case, convergence

to the reference solution can be observed as the order of the shape functions are

increased. However, a sudden jump in the strain energy can be observed at around 5µs

while using quadratic ansatz. This could be due to the insufficient time step resolution

for the given discretization.

Remark

The same case is now solved using the hybrid formulation. The results for crack

propagation are shown in Fig.4.27. Two stagegred iterations could be insufficicent in

the case of the hybrid formulation to achieve equilibrium between the elastic and the

phase-field problems. However, this needs further investigation and is out of the scope

of this work.

p=

1

k=

5

p=

2

k=

5

Hybrid formulation

(a) t = 20µs (b) t = 50µs (c) t = 80µs


refinement for different ansatz order with k = 5 using the hybrid formula-

tion. Crack phase-field evolution over time.

58

5 Summary and outlook

5.1 Summary

The phase-field model for quasi-static and dynamic brittle fracture has successfully been

integrated with a uniform multi-level hp-adaptive refinement scheme and the Finite

Cell Method for both the anisotropic Miehe and the hybrid formulations. Validation

of the implementation has been done through benchmark examples. A heuristic

refinement criteria to refine the mesh dynamically (over the course of the simulation)

has been developed to ensure h-refinement around the region with high gradients due

to the presence of crack. Taking advantage of phase-field models where the idea is to

have a diffusive crack which eliminates the necessity of algorithmic tracking of crack,

the heuristic criteria remains numerically simple and robust. Having developed the

adaptive refinement criteria, the benefits obtained from a local refinement scheme over

a global refinement one has been clearly established using the single-edge notched

shear test example. Importance of the number of staggered iterations used in solving

the two initial boundary value problems using a staggered approach has been studied

in the notched plate with hole example which also demonstrates the application of the

FCM for phase-field models. The dynamic uniaxial tension example is an evidence for

the robustness of the refinement criteria.

5.2 Outlook

The research in this work is not bounded. Undoubtedly, it provides a proof for the

applicability of the phase-field model in the context of uniform multi-level hp-adaptive

refinement and the Finite Cell Method but has also raised open questions. Following

addresses the same.

In the uniform multi-level hp-refinement scheme, the polynomial order of the

overlay meshes remain the same as the base mesh. This scheme has been

used in this work due to its robustness. However, crack has been studied as

a diffusive phenomenon in the phase-field model where the stress degradation

term is quadratic. Hence, it would be ideal to have a non-uniform hp-refinement

where the polynomial order is graded over the overlay mesh. This is done in

the non-uniform multi-level hp-refinement scheme. Implementation of such a

scenario remains open.

59

5 Summary and outlook

In the example where a concrete block subjected to compression is studied, the

behavior of the two formulations are not completely understood. This needs

further insight about the applicability of the formulations in such scenarios.

The results for dynamic crack branching using the hybrid formulations are not

the same as that from the anisotropic Miehe formulation. A further investigation

needs to be done in this regard.

In this work, a second-order phase-field theory with second derivative of the

phase-field has been applied in the strong form governing the fracture process.

Fourth-order theories with higher derivatives have been proposed to achieve

higher convergence rate when used with smooth bases. Hence, extension of this

work to a fourth order theory along with isogeometric framework could prove

beneficial.

Extension to three-dimensional problems, combining the elegant modeling capabil-

ities of phase-field models along with the above studied multi-level hp-refinement

technique to dynamically and locally refine the mesh only in the zone of interest

is a potential research area.

60

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