Continuous-discontinuous modelling of failure based on non

Universitat Politècnica de Catalunya
Programa de Doctorat de Matemàtica Aplicada
Departament de Matemàtica Aplicada III
Continuous-discontinuous modelling of failure
based on non-local displacements
by
Elena Tamayo-Mas
Thesis Proposal
Advisor: Antonio Rodrı́guez-Ferran
Barcelona, July 2010
Abstract
Continuous-discontinuous modelling of failure based on non-local
displacements
Elena Tamayo-Mas
Two different kinds of approaches are typically used to model failure of quasibrittle materials: (a) damage mechanics and (b) fracture mechanics. The former,
which belongs to the family of continuous models, is able to capture damage inception
and its diffuse propagation. The latter, which falls in the family of discontinuous
models, can be used to model the final stages of failure processes, when the body is
physically separated in two or more parts. In order to achieve a better description of
the entire failure process, integrated strategies which combine these two traditional
approaches have recently emerged. In this work, a new contribution in this direction
is presented: a non-local continuum damage model based on non-local displacements
is used to simulate the initial stages of failure whereas it is coupled to a discontinuous
model to capture crack initiation and its propagation.
iii
Contents
Abstract
iii
Contents
v
List of Figures
vii
List of Tables
xi
List of Symbols
xiii
Latin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Greek symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Layout of the proposal . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 State of the art
2.1 Continuum failure models . . . . . . . . . . . . . . . . .
2.1.1 Continuum damage models . . . . . . . . . . . .
2.2 Discontinuous failure models . . . . . . . . . . . . . . . .
2.2.1 Cohesive crack models . . . . . . . . . . . . . . .
2.2.2 Computational modelling of strong discontinuities
2.3 Continuous-discontinuous failure models . . . . . . . . .
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3 Continuous model of failure based on non-local displacements
3.1 Gradient continuous non-local damage model . . . . . . . . . . . .
3.2 Boundary conditions for the regularisation equation . . . . . . . .
3.3 Numerical examples: validation of the model . . . . . . . . . . . .
3.3.1 Square plate under mode I loading conditions . . . . . . .
3.3.2 Single-edge notched beam . . . . . . . . . . . . . . . . . .
3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Continuous-discontinuous model based on non-local displacements
4.1 Gradient continuous-discontinuous non-local damage model . . . . . . .
4.1.1 Problem fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . .
4.1.4 Discretised and linearised weak governing equations . . . . . . .
4.2 Finite element technology . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Introducing a discontinuity . . . . . . . . . . . . . . . . . . . . .
4.2.2 Orienting a discontinuity . . . . . . . . . . . . . . . . . . . . . .
4.3 Numerical examples: validation of the model . . . . . . . . . . . . . . .
4.3.1 2D uniaxial tension test . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Square plate under mode I loading conditions . . . . . . . . . .
4.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
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50
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51
52
52
54
57
5 Future work
61
6 Publications
65
Bibliography
71
vi
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.1
3.2
3.3
3.4
3.5
Kinematic description with (a) strong discontinuity; (b) weak discontinuity and (c) no discontinuities. . . . . . . . . . . . . . . . . . . . . .
Cohesive crack process zone. . . . . . . . . . . . . . . . . . . . . . . . .
Different cohesive laws used in the literature: (a) linear with secant
unloading (b) bilinear and (c) exponential laws. . . . . . . . . . . . . .
The three fracture modes: (a) Mode I or opening mode (b) Mode II or
sliding mode and (c) Mode III or tearing mode. . . . . . . . . . . . . .
The mesh conforms to crack geometry. . . . . . . . . . . . . . . . . . .
Triangular element crossed by a crack (embedded discontinuity model).
A crack line (dashed line) in a structured mesh with standard elements
(white), elements whose nodes are all enriched (dark grey) and blending
elements (light grey). Nodes in J and K are indicated by circles and
squares respectively. Adapted from Moës et al. (1999). . . . . . . . . .
Polar coordinates used to evaluate the tip enrichment functions. . . . .
Subdomain quadrature: the elements cut by the crack are subdivided
into subdomains (dashed lines). . . . . . . . . . . . . . . . . . . . . . .
x) = ua (x, y) = 1 +
Validation test with (a) a linear source term ua (x
x + 5y. Solutions obtained by means of (b) Dirichlet, (c) Neumann, (d)
Combined (homogeneous Neumann) and (e) Combined (non-homogeneous
Neumann) boundary conditions. . . . . . . . . . . . . . . . . . . . . . .
Validation test with (a) a tent function source term. Solutions obtained
by means of (b) Dirichlet, (c) Neumann, (d) Combined (homogeneous
Neumann) and (e) Combined (non-homogeneous Neumann) boundary
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Validation test with (a) a tent function source term. Solutions obtained
by means of (b) Dirichlet, (c) Neumann, (d) Combined (homogeneous
Neumann) and (e) Combined (non-homogeneous Neumann) boundary
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Square plate under mode I loading conditions: problem statement. . . .
Square plate under mode I loading conditions. Four meshes with different element density and different imperfection sizes are used. . . . .
vii
10
11
11
12
13
14
16
17
18
26
27
28
29
30
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
Square plate under mode I loading conditions. Fixed characteristic
length with various meshes and imperfection sizes: damage profiles
with deformed meshes (×100). . . . . . . . . . . . . . . . . . . . . . .
Square plate under mode I loading conditions. Fixed characteristic
length with various meshes and imperfection sizes: force-displacement
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Single-edge notched beam: problem statement. All distances in mm. .
SENB reference test. Four meshes with different element density are
used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
SENB reference test. Fixed characteristic length ` = 10 mm with
various meshes: final damage profiles. . . . . . . . . . . . . . . . . . .
√
SENB reference test. Fixed characteristic length ` = 10 mm with
various meshes: force-displacement curves. . . . . . . . . . . . . . . .
SENB reference test. Fixed characteristic length ` = 10 mm with
various meshes: force-displacement curves. . . . . . . . . . . . . . . .
SENB reference test. Fixed characteristic length ` = 10 mm with
various meshes: final damage profiles. . . . . . . . . . . . . . . . . . .
. 31
. 32
. 32
. 36
. 37
. 38
. 38
. 39
4.1
Notations for a body with a crack subjected to loads and imposed
displacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Constitutive law for the cracked bulk: (a) elastic unloading with D =
Dcrit ; (b) linear traction-separation law for the crack. . . . . . . . . . .
4.3 Sketch of the cohesive tip region. A crack line (dashed line) in a mesh
with standard elements (white) and elements whose nodes are enriched
(grey). Enriched nodes with H = +1 and H = −1 are indicated by
squares and circles respectively. . . . . . . . . . . . . . . . . . . . . . .
4.4 Determination of the propagation direction: the crack propagates according to the direction perpendicular to the isolines of the damage
field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Uniaxial tension test: problem statement. . . . . . . . . . . . . . . . . .
4.6 Evolution law for (a) the bar; (b) the crack. . . . . . . . . . . . . . . .
√
4.7 Uniaxial tension test. Fixed characteristic length ` = 5 with various
meshes. (a) force-displacement curve; (b) damage profiles. . . . . . . .
4.8 Uniaxial tension test. Fixed mesh with various characteristic lengths.
(a) force-displacement curve; (b) damage profiles. . . . . . . . . . . . .
4.9 Uniaxial tension test. Fixed mesh and characteristic length with various imperfection sizes. (a) force-displacement curve; (b) damage profiles.
4.10 Square plate under mode I loading conditions. Fixed characteristic
length with various meshes and imperfection sizes: damage profiles
with deformed meshes (×100). . . . . . . . . . . . . . . . . . . . . . . .
viii
42
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51
52
53
53
54
55
55
56
4.11 Square plate under mode I loading conditions. Fixed characteristic
length with various meshes and imperfection sizes: force-displacement
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1
5.2
Work schedule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Representation of the branching. . . . . . . . . . . . . . . . . . . . . . . 63
ix
List of Tables
2.1
2.2
2.3
Generic equations of a local damage model. . . . . . . . . . . . . . . .
Generic equations of an integral-type non-local damage model. . . . . .
Generic equations of a gradient-type non-local damage model. . . . . .
3.1
3.2
Damage model based on non-local displacements, gradient version. . . . 22
Square plate under mode I loading conditions: geometric and material
parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
SENB reference test: material parameters for concrete beam (modified
von Mises model with exponential damage evolution) and steel loading
plates (elastic model). . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3
4.1
4.2
9
9
9
Matrices belonging to the consistent tangent matrix. . . . . . . . . . . 49
Uniaxial tension test with a non-local damage model and a softening
behaviour of the cohesive crack: geometric and material parameters. . . 53
xi
List of Symbols
Latin symbols
a
A
b
B
B
c
C
D
Dcrit
D
DHΓd
H
I1
J2
k
`
M
MHΓd
n
Ni
N
r
t
t̄
t̄ d
t
tn
ts
Vector of standard nodal displacement degrees of freedom
Parameter which controls the residual strength in a damage model
Vector of step enriched nodal displacement degrees of freedom
Parameter which controls the slope of the softening branch at the peak
in a damage model
Derivative shape function matrix
Vector of tip enriched nodal displacement degrees of freedom
Tensor of elastic moduli
Damage parameter
Critical damage value
Diffusivity matrix
Enriched diffusivity matrix
Heaviside function
First invariant of the strain tensor
Second invariant of the deviatoric strain tensor
Ratio of compressive strength to tensile strength in the modified von
Mises model
Characteristic length of a non-local damage model
Mass matrix
Enriched mass matrix
Outward unit normal to the boundary
Shape function of node i
Shape function matrix
Radial coordinate at the crack tip
Time
Traction vector
Load on the discontinuity surface
Unit tangent to the boundary
Normal traction
Shear traction
xiii
T
u
un
ut
u1
u2
ua
u1a
u 2a
ug
u 1g
u 2g
u
ua
u1a
u2a
ug
u1g
u2g
uK
Ju
JuKn
JuKs
x
Y
Y0
Yf
Ye
Yg
Tangent matrix for the discontinuity
Displacement field
Normal component of the displacement jump (embedded discontinuity
models)
Tangential component of the displacement jump (embedded discontinuity models)
Standard displacement
Enhanced displacement
Local displacement field in the damage model based on non-local
displacements
Component of ua when Ω̄ is crossed by Γd
Component of u a when Ω̄ is crossed by Γd
Non-local displacement field in the damage model based on non-local
displacements
Component of u g when Ω̄ is crossed by Γd
Component of u g when Ω̄ is crossed by Γd
Nodal displacements
Local nodal displacements
Component of ua when Ω̄ is crossed by Γd
Component of ua when Ω̄ is crossed by Γd
Non-local nodal displacements
Component of ug when Ω̄ is crossed by Γd
Component of ug when Ω̄ is crossed by Γd
Discrete displacement jump
Crack opening
Sliding of the crack faces
Spatial coordinate vector
Local state variable of a damage model
Damage threshold
Maximum admissible value for the state variable
Non-local state variable of a damage model
Non-local state variable of a damage model (via non-local
displacements)
Greek symbols
α
Γ
Γd
Weighting function of an integral-type non-local model
Boundary surface
Discontinuity surface
xiv
Γt
Γu
δΓ d
ε
εg
εi
θ
ν
σ
ψ
Ω
Boundary with prescribed tractions
Boundary with prescribed displacements
Dirac-delta distribution centred at the discontinuity Γd
Small strain tensor
Small strain tensor (via non-local displacements)
Principal strains
Angular coordinate at the crack tip
Poisson’s coefficient
Cauchy stress tensor
Generic enrichment function (X-FEM)
Domain
Operators
:
Double contraction
∇
Nabla operator
∇2
Laplace operator
∇s
Symmetrised gradient
xv
Chapter 1
Introduction
1.1
Motivation
Failure of quasi-brittle materials is a process characterised by formation of microcracks, strain localisation and the accumulation of damage, thus leading to the possible
development of macrocracks. As a consequence, when modelling failure phenomena,
all these features must be taken into account: the numerical strategy should consider
the inception and the propagation of damage, the possible evolution of microcracks
into macrocracks and the correct macrocrack-microcrack interaction.
To simulate numerically a failure process, two different kinds of approaches have
usually been employed: (a) damage mechanics, which belongs to the family of continuous models and (b) fracture mechanics, which belongs to the discontinuous models.
On the one hand, by means of damage mechanics analyses, the first stages of
failure of quasi-brittle materials may be described. These models are characterised
by a strain softening phenomenon, which does not correctly reflect the energy dissipated in the fracture process zone. Therefore, if stress-strain laws with softening are
used, numerical simulations present a pathological mesh sensitivity thus leading to
physically unrealistic results. To solve the pathological mesh dependence, a regularisation technique may be used to incorporate non-locality into the model, either by an
integral-type model or a gradient-type approach. However, despite the regularisation,
1
2
Introduction
non-local continuum damage models cannot be used in the final stage of failure, when
the body is physically separated in two or more parts. Since in these models the body
is treated as a continuum body, numerical interaction between the separated parts of
the body persists and unrealistic results may be obtained.
On the other hand, fracture mechanics analyses, based on the cohesive zone concept, are able to deal with discontinuities. Hence, they can be employed in order to
capture the last stages of failure without formation of spurious damage growth. From
a numerical viewpoint, their applications were first restricted, since the standard finite
element method requires that cracks must propagate along element boundaries. Nevertheless, different methods such as the eXtended Finite Element Method (X-FEM)
have emerged in order to overcome this limitation and nowadays discontinuous models
can adequately be used in the final stages of failure. However, discontinuous models
are not able to describe neither damage inception nor its diffuse propagation.
As suggested by the above discussion, in order to achieve a better characterisation
of the whole failure process, a new kind of model which combine these two traditional
approaches have recently appeared. The basic idea of these continuous-discontinuous
strategies is to employ damage mechanics in order to describe the inception and
the propagation of damage and fracture mechanics in order to deal with cracks and
material separation.
In this proposal, a new continuous-discontinuous approach is presented. A regularised continuum model based on non-local displacements is coupled to a discontinuous model when the damage parameter exceeds a threshold set a priori. Then, a crack
described by a cohesive law is introduced and propagated through the continuous bulk
according to the direction dictated by the continuum.
1.2
Layout of the proposal
This proposal is organised as follows.
Chapter 2 exposes the present state of the art in failure modelling by means of
finite element techniques: Section 2.1 focuses on the continuum failure models, Section
1.2 Layout of the proposal
3
2.2 details the discrete approaches and finally, Section 2.3 describes some continuousdiscontinuous techniques already presented in the literature.
Chapter 3 presents the candidate’s work on the regularised approach in order to
achieve a realistic description of the first stages of failure phenomena. It is structured
in the following sections: in Section 3.1 the gradient version of the damage model based
on non-local displacements is briefly reviewed, Section 3.2 deals with the definition of
the boundary conditions for the regularisation equation and some numerical examples
to illustrate the regularisation capabilities are presented in Section 3.3.
In Chapter 4, the coupling between the continuous and the discontinuous approach is explained. In Section 4.1, the introduction of discontinuities in the implicit
gradient-enhanced continuum model based on non-local displacements is presented.
Special emphasis is placed on some issues pertinent to the implementation of the
combined strategy in Section 4.2. The regularisation capabilities of this continuousdiscontinuous approach are illustrated in Section 4.3 by means of two numerical examples.
Finally, in Chapter 5, the future work to be developed is described and a tentative
calendar is proposed and in Chapter 6, the list of congress participations is detailed.
Chapter 2
State of the art
In the present state of the art in finite element techniques, two different approaches to
model failure phenomena are considered. On the one hand, continuum strategies can
be used, where the fracture is conceived as the result of a process of strain localisation
and damage accumulation. Therefore, there is not any real discontinuity in the material. On the other hand, discrete approaches may be employed, where discontinuities
are introduced into the model.
In this chapter, these two kinds of techniques are reviewed. First, in Section 2.1,
the continuum failure models and the need for its regularisation are considered. Damage models are explained in detail in Section 2.1.1. In Section 2.2, an overview of discrete approaches is given. On the one hand, in Section 2.2.1, some general comments
on traction-separation laws are made. On the other hand, in Section 2.2.2, the main
discretisation methods are discussed. To finish with, some continuous-discontinuous
techniques are presented in Section 2.3.
2.1
Continuum failure models
In quasi-brittle materials subjected to loading conditions, the stress-strain curve is
nearly linear up to the peak stress, whereas it immediately decreases after it. This
phenomenon, which is known as strain softening, leads to a physically unrealistic
5
6
State of the art
treatment of the energy dissipated during the failure process. Regarding numerical
simulations, if stress-strain laws with softening are used, the results present a pathological mesh sensitivity: as the finite element mesh employed in the simulations is
refined, the energy dissipated in the fracture process tends to zero (Jirásek, 2007a).
Different solutions have been proposed in the literature to remedy this physically
unrealistic behaviour (Jirásek, 2007b):
• By means of the cohesive crack model, (Hillerborg et al., 1976), which admits
jumps in the displacement field, softening is described by a traction-separation
law. Hence, tractions transmitted by the crack are related to the displacement
jump (Oliver et al., 2002).
• In the crack band model, (Bažant and Oh, 1983), the process zone is represented
by a band of highly localised strain. Since in numerical computations the width
of this band is controlled by the size of finite elements, the softening modulus is
defined according to the finite element size (Cervera and Chiumenti, 2006).
• Regularised models, which consist of introducing a non-local effect, incorporate
a material characteristic length that prevents strain localisation into a line thus
leading to zero energy dissipation (Comi, 2001). Examples of these strategies
include non-local integral or gradient-enriched formulations. On the one hand, in
integral-type models, a non-local state variable is defined as the weighted average
of the local state variable in a neighbourhood of the point under consideration
(Bažant and Jirásek, 2002). On the other hand, in gradient-type models, higherorder derivatives are added to the partial differential equation that describes the
evolution of the non-local variable (de Borst et al., 1995).
In this proposal, we focus on this technique.
2.1.1
Continuum damage models
Continuous damage models may be employed to describe the evolution of failure
processes, between the undamaged state and macroscopic crack initiation (Lemaitre
2.1 Continuum failure models
7
and Chaboche, 1990). A generic local damage model (Table 2.1) is characterised by
the following equations:
• Constitutive equation
x, t) = (1 − D (x
x, t)) C : ε (x
x, t)
σ (x
(2.1)
where σ is the Cauchy stress tensor, ε the small strain tensor, C the tensor of
elastic moduli, D the damage parameter, which ranges between 0 (undamaged
material) and 1 (completely damaged material) and (:) denotes the double
contraction.
• Strains
x, t) = ∇su (x
x, t)
ε (x
(2.2)
where ∇s is the symmetrised gradient and u stands for the displacement field.
• Local state variable
It is assumed that the local state variable Y depends on the strains.
x, t) = Y (εε (x
x, t))
Y (x
(2.3)
Y should account for those features of the strain field which are responsible
for damage inception and propagation. Moreover, Y should be more sensitive
to positive than to negative strains. In the Mazars model (Mazars, 1986), for
example,
Y =
s
X
[max (0, εi )]2
(2.4)
i
where εi are the principal strains.
On the other hand, in the modified von Mises model (de Vree et al., 1995),
k−1
1
Y =
I1 +
2k (1 − 2ν)
2k
s
k−1
I1
1 − 2ν
2
+
12k
J2
(1 + ν)2
(2.5)
8
State of the art
where k is the ratio of compressive strength to tensile strength, ν is the Poisson’s
coefficient, I1 is the first invariant of the strain tensor and J2 is the second
invariant of the deviatoric strain tensor.
• Damage evolution
It is assumed that the damage parameter D depends on the state variable Y .
x, t) = D (Y (x
x, t))
D (x
(2.6)
In damage models, D starts above a threshold Y0 (D = 0 for Y ≤ Y0 ) and
cannot decrease (Ḋ ≥ 0). The most common expressions are the following:
– Exponential law
Y0 (1 − A)
− Ae−B(Y −Y0 )
Y
(2.7)
1
1 + B (Y − Y0 ) + A (Y − Y0 )2
(2.8)
D =1−
– Polynomial law
D =1−
where the parameters A and B control the residual strength and the slope
of the softening branch at the peak respectively.
– Linear softening branch
Yf
D=
Yf − Y0
Y0
1−
Y
(2.9)
where Yf is the maximum admissible value for the state variable.
In the literature, various articles that deal particularly with damage models and the
different ways to fix their spurious mesh sensitivity, either via integral-type (PijaudierCabot and Bažant, 1987) or gradient-type models (Peerlings et al., 1998), can be
found. In standard regularised damage models, integral-type (Table 2.2) or gradient-
2.1 Continuum failure models
9
Table 2.1: Generic equations of a local damage model.
Constitutive equation
Strains
Local state variable
Damage evolution
x, t) = (1 − D (x
x, t)) C : ε (x
x, t)
σ (x
x, t) = ∇su (x
x, t)
ε (x
x, t) = Y (εε (x
x, t))
Y (x
x, t) = D (Y (x
x, t))
D (x
(2.10a)
(2.10b)
(2.10c)
(2.10d)
type (Table 2.3), the state variable Y incorporates the non-local effect via a characteristic length `.
Table 2.2: Generic equations of an integral-type non-local damage model.
Constitutive equation
Strains
Local state variable
Non-local state variable
Damage evolution
x, t) = (1 − D (x
x, t)) C : ε (x
x, t) (2.11a)
σ (x
s
x, t) = ∇ u (x
x, t)
ε (x
(2.11b)
x, t) = Y (εε (x
x, t))
Y (x
(2.11c)
R
α (d) Y dV
x, t) = RV
Ye (x
(2.11d)
α
(d)
dV
V
x, t) = D(Ye )
D (x
(2.11e)
Table 2.3: Generic equations of a gradient-type non-local damage model.
Constitutive equation
Strains
Local state variable
Non-local state variable
Damage evolution
x, t) = (1 − D (x
x, t)) C : ε (x
x, t) (2.12a)
σ (x
s
x, t) = ∇ u (x
x, t)
ε (x
(2.12b)
x, t) = Y (εε (x
x, t))
Y (x
(2.12c)
2 2e
e
x, t) − ` ∇ Y (x
x, t) = Y (x
x, t) (2.12d)
Y (x
x, t) = D(Ye )
D (x
(2.12e)
10
State of the art
On the one hand, in Eq. (2.11d) the non-locality is introduced by means of a
weighting function α, which depends on the distance d to the point under consideration
and contains the characteristic length ` as a parameter. On the other hand, in Eq.
(2.12d), Ye is the solution of a partial differential equation where Y is the source term.
As shown in Jirásek (1998), apart from the state variable Y , different variables
can be employed to add non-locality into the model. In this work, we propose to use a
gradient non-local model based on non-local displacements (Rodrı́guez-Ferran et al.,
2005) to regularise softening.
2.2
Discontinuous failure models
In materials subjected to very extreme loading conditions, the localisation of deformation provokes that some macroscopical discontinuities arise. These discontinuities may
be modelled as jumps in the displacement field and are named strong discontinuities,
see Figure 2.1.
(a)
(b)
(c)
Figure 2.1: Kinematic description with (a) strong discontinuity; (b) weak discontinuity
and (c) no discontinuities.
2.2 Discontinuous failure models
2.2.1
11
Cohesive crack models
In contrast to continuum models, with smeared cracks, discontinuous models incorporate discontinuous displacement fields, thus leading to the necessity of dealing with
formation and growth of cracks. In order to characterise these propagating discontinuities, different techniques, mainly based on the cohesive crack concept (Hillerborg
et al., 1976), have been developed. In these approaches, the fracture is seen as a
gradual phenomenon: separation occurs across an extended crack tip or a cohesive
zone, see Figure 2.2.
Figure 2.2: Cohesive crack process zone.
To model this behaviour, a special type of constitutive law describing the crack
is used. Therefore, two different kinds of constitutive laws are employed: (a) the
usual stress-strain law describing the bulk material, which is still continuous and (b)
a traction-separation law characterising the crack. In Figure 2.3, three different kinds
of cohesive laws for one-dimensional problems are shown.
(a)
(b)
(c)
Figure 2.3: Different cohesive laws used in the literature: (a) linear with secant
unloading (b) bilinear and (c) exponential laws.
12
State of the art
This traction-separation law, whose definition depends on the mode of fracture
(Figure 2.4), relates the tractions t̄ = [tn , ts ] transmitted by the crack to the disuK = [JuKn , JuKs ], where JuKn is the crack opening and JuKs is the
placement jump Ju
relative sliding of the crack faces.
(a)
(b)
(c)
Figure 2.4: The three fracture modes: (a) Mode I or opening mode (b) Mode II or
sliding mode and (c) Mode III or tearing mode.
2.2.2
Computational modelling of strong discontinuities
The Finite Element Method (FEM), which performs well approximating smooth functions, is not suited for the approximation of non-smooth solutions. Hence, special
techniques (Jirásek and Belytschko, 2002) have to be used to incorporate displacement discontinuities in numerical models. Here, these methods are briefly described.
Remeshing
In remeshing methods (Bouchard et al., 2000, 2003; Patzák and Jirásek, 2004), the
standard FEM is used. Nevertheless, an appropriate mesh must be built: in these
methods, element faces (or element edges in two dimensions) have to be aligned with
the crack and its nodes are doubled and distributed to both sides of the discontinuity,
see Figure 2.5.
One advantage of this method is its easy implementation, since only a mesher and
a standard finite element program are needed. Nevertheless, remeshing as the crack
2.2 Discontinuous failure models
13
Figure 2.5: The mesh conforms to crack geometry.
grows presents certain limitations:
• Remeshing requires projection of variables between the different meshes and
causes some difficulties during post-processing.
• Mesh refinement is required where the solution is expected to have singularities,
thus leading to an expensive technique from the computational viewpoint.
• In some problems dealing with non-linear crack propagation or the growth of
intersecting cracks, having an adequate mesh is very difficult or even impossible.
Due to these disadvantages, other methods allowing cracks with arbitrary trajectory have been developed.
Embedded discontinuities
In embedded discontinuity models (see Jirásek (2000) for classification), a discrete
crack is modelled through the inclusion of strong discontinuities embedded in finite
elements. In this formulation, the cracks are not necessarily aligned with the element
boundaries but they are able to grow across the continuum elements.
In embedded discontinuity models, the displacement field can be decomposed into
a continuous and a discontinuous part. The crack is then represented by additional
degrees of freedom referring to the normal un and tangential ut component of the
displacement jump, see Figure 2.6(b).
14
State of the art
(a)
(b)
Figure 2.6: Triangular element crossed by a crack (embedded discontinuity model).
Elements with embedded discontinuities provide better approximation solutions
than pure continuum models, which smear displacement jumps. On the other hand,
compared to remeshing techniques, here the entire crack may be represented independently of the finite element mesh. Moreover, since the additional degrees of freedom
un and ut have an internal character, no extra global degrees of freedom are introduced and the global equilibrium equations may be written only in terms of the nodal
displacements. However, as discussed in Jirásek and Belytschko (2002), elements with
embedded discontinuities still present some limitations. By means of this formulation,
it is possible to reproduce an arbitrary jump at a certain point, but in the split element, strains on both sides of the crack remain the same since the strain in the bulk
material is still interpolated in a continuous way.
Due to this disadvantage, it is preferable to use the eXtended Finite Element
Method (X-FEM), which is able to reproduce exactly the separation and the independent deformation of the split parts.
Extended finite element method
One of the latest improvements in modelling crack growth are the methods based
on the partition of unity concept (Melenk and Babuška, 1996; Babuška and Melenk, 1998), among which the eXtended Finite Element Method (X-FEM) is the most
employed. Detailed overviews of the developments of X-FEM have been given by
Karihaloo and Xiao (2003); Abdelaziz and Hamouine (2008); Yazid et al. (2009); Belytschko et al. (2009). Here, the basic features of crack growing by means of X-FEM
2.2 Discontinuous failure models
15
are briefly reviewed.
Main features of X-FEM In X-FEM (Belytschko and Black, 1999; Moës et al.,
1999), displacements are approximated by the sum of two continuous displacement
fields, one of which is multiplied by a discontinuous function in order to model the discontinuities. Thus, the continuous part corresponds to the displacement field without
any crack, while the discontinuous or the enriched contribution (the partition of unity
enrichment) takes into account the discontinuities. Hence, the displacement field u
can be decomposed as
x) = u 1 (x
x) + ψ(x
x)u
u2 (x
x)
u (x
(2.13)
where u 1 and u 2 , which are continuous fields, are the regular and enhanced displacex) is the enrichment function, which is discontinuous
ment fields respectively and ψ(x
across the domain.
Therefore, the basic idea of X-FEM is to add discontinuous enrichment functions
to the finite element approximation in order to include information from asymptotic
solutions and other knowledge of the properties of the solution. The enrichment
function is then chosen depending on the kind of information which is incorporated
into the solution. For the particular case of crack modelling, X-FEM should take into
account two kinds of aspects: (a) the discontinuity field along the crack faces and (b)
the crack tip singularity.
Enrichment functions Consider a finite element mesh of a cracked body, as seen
in Figure 2.7.
Let I denote the set of all nodes in the finite element mesh, J the set of nodes
of elements around the crack tip (called tip enriched nodes and indicated by circles
in Figure 2.7) and K the set of nodes of elements crossed by the crack but not in J
(called step enriched nodes and indicated by squares in Figure 2.7). Then, the X-FEM
displacement approximation for a crack is
!
x) =
u(x
X
i∈I
x)ai +
Ni (x
X
j∈J
x)H(x
x)bj +
Nj (x
X
k∈K
x)
Nk (x
X
l
x)clk
ψ l (x
(2.14)
16
State of the art
Figure 2.7: A crack line (dashed line) in a structured mesh with standard elements
(white), elements whose nodes are all enriched (dark grey) and blending elements
(light grey). Nodes in J and K are indicated by circles and squares respectively.
Adapted from Moës et al. (1999).
x) is the Heaviside step function
where Ni are the standard FEM shape functions, H(x
x) is a
which allows to represent the displacement jump along the crack faces, ψ l (x
set of enrichment functions which approximates the crack tip behaviour, ai are the
standard nodal degrees of freedom, bi are the step enriched nodal degrees of freedom
and ci are the tip enriched nodal degrees of freedom.
To approximate the crack tip behaviour, Belytschko and Black (1999); Stolarska
et al. (2001) use for the two-dimensional problem the functions
√
θ
θ
θ
θ
ψ (r, θ) = r sin , cos , sin sin θ, cos sin θ
2
2
2
2
l
(2.15)
where r and θ are polar coordinates with origin at the crack tip and parallel to the
crack faces, see Figure 2.8.
To approximate the displacement jump along the crack faces, different step functions have been used in the literature, such as the Heaviside (Simone et al., 2003;
Wells et al., 2002; Wells and Sluys, 2001) or the sign function (Moës and Belytschko,
2002; Zi and Belytschko, 2003; Comi and Mariani, 2007; Bechet et al., 2005; Mariani
and Perego, 2003).
2.2 Discontinuous failure models
17
Figure 2.8: Polar coordinates used to evaluate the tip enrichment functions.
It should be stressed that different authors (see for example Comi et al. (2007);
Simone et al. (2003)) assume that the crack tip always belongs to an element edge
thus avoiding the use of tip enrichment functions. Then, the X-FEM displacement
approximation for a crack is simply
x) =
u(x
X
i∈I
x)ai +
Ni (x
X
x)H(x
x)bj
Nj (x
(2.16)
j∈J
Note that X-FEM enrichment involves extra degrees of freedom for the nodes in
the domain subjected to the enrichment. Nevertheless, in Eq. (2.14) and (2.16),
the enrichment is added only where it is needed thus leading to a computationally
efficient method since in general few extra unknowns are added. Due to this local
enrichment, three different kinds of elements exist: (a) standard elements with no
enriched nodes, (b) elements whose nodes are all enriched and (c) elements with some
of their nodes enriched, see Figure 2.7. These hybrid elements are commonly named
blending elements and involve problems in the solution accuracy as reported in Fries
(2008); Chessa et al. (2003); Gracie et al. (2008).
Numerical integration When integrating X-FEM functions, two different difficulties arise: both the discontinuous functions along the crack faces and the singularities
at the crack tip must be taken into account. Since the traditional quadrature rules,
for example Gauss quadratures, are designed to integrate polynomials and functions
that are well approximated by polynomials, these quadratures are not valid to inte-
18
State of the art
grate X-FEM functions properly. In order to do that, several different approaches
have been used in the literature, see Belytschko et al. (2009).
The usual method is to employ subdomain quadratures, see Belytschko and Black
(1999); Moës et al. (1999), in which the element is subdivided into quadrature subdomains, whose boundaries are aligned with the crack, see Figure 2.9. For elements cut
by the crack and enriched with the step function, this strategy solves the problem,
since the functions are continuous in each subdomain and standard quadratures may
be employed. Nevertheless, for elements containing the crack tip, there still exists
a singularity in the displacement field. In standard FEM, two different techniques
may be used to solve problems dealing with singularities: (a) increasing the number
of integration points, which does not perform well with X-FEM, see Zlotnik (2008),
and (b) using special remeshing techniques that we want to avoid. In order to solve
this problem, special methods, consisting also of subdivisions, have been proposed in
Laborde et al. (2005) and Bechet et al. (2005).
Figure 2.9: Subdomain quadrature: the elements cut by the crack are subdivided into
subdomains (dashed lines).
2.3
Continuous-discontinuous failure models
Continuous-discontinuous models emerged to achieve a better characterisation of the
whole failure process. The basic idea of these integrated strategies is to combine
continuous and discontinuous models. The former obtain realistic results in the first
stages of failure process, while are not able to reproduce cracks and material separa-
2.3 Continuous-discontinuous failure models
19
tion. On the other hand, discontinuous models can describe the propagating cracks
but not the initial states of fracture (Mazars and Pijaudier-Cabot, 1996). The main
features of these combined strategies are summarised here:
• Non-local continuous models are used to describe the first stages of failure. Thus,
numerical simulations do not present mesh sensitivity and physically realistic
results are obtained.
• At the end of each time step, the approach checks if the transition criterion is
fulfilled. In such a case, a discrete cohesive crack is introduced into the model
and the direction of its propagation is determined.
• From that moment on, a discontinuous approach is used to model the final
stages of the failure process. Therefore, the numerical interaction between the
separated parts of the body ceases and realistic results may be obtained.
Different integrated strategies have been already proposed in the literature, some of
which are briefly reviewed here.
Jirásek and Zimmermann (2001) analyse the combination of the smeared and
embedded descriptions of cracking. The authors propose to use a model dealing with
inelastic strain in order to characterise the early stages of material degradation and
displacement discontinuities to deal with the highly localised fracture.
In Wells et al. (2002), a numerical model that allows traction-free discontinuities
in a softening viscoplastic bulk is presented. Here, the transition takes place when
a critical threshold of inelastic deformation is reached. From that moment on, when
the plastic deformation in the continuum around a discontinuity tip reaches a critical
level, the discontinuous surface is extended.
In Simone et al. (2003), an implicit gradient-enhanced continuum damage model is
used to describe the first steps of the fracture process. Nevertheless, when the damage
parameter is close to one, a traction-free crack is introduced and a discontinuous
interpolation of the problem fields is employed thus preventing from spurious growth
of damage.
20
State of the art
In Comi et al. (2007), the early steps of failure process are also modelled by means
of a non-local continuum damage approach. When the damage parameter exceeds a
critical value, which is defined a priori, this is combined to an extended finite element
cohesive crack model in order to describe the final stages of failure. The transition
from the continuous to the discontinuous approach is based on an energy balance,
that is the fracture energy not yet dissipated in the damage band is transferred to the
cohesive zone, while the bulk unloads.
In this work, a new contribution in this direction is presented. A non-local continuum damage model based on non-local displacements is used for the continuum.
When the damage parameter exceeds a threshold set a priori Dcrit , a crack described
by a cohesive law relating traction to displacement jump is introduced. We propose to
determine the crack path by means of the continuum: the gradient of the damage field
is employed. Once the crack is introduced, the growing crack is modelled by means of
the X-FEM. In this model, the transition from a continuum to a discontinuous model
is defined as follows: damage value is fixed to Dcrit and the bulk material unloads.
Chapter 3
Continuous model of failure based
on non-local displacements
Regularised damage formulations provide an objective description of the first stages
of a failure process, when the bulk is considered to be continuous. They are based
on non-local fields which incorporate non-locality into the model. In this work, a
formulation with regularised displacements is employed in order to obtain physically
realistic results. The idea of a damage model with non-local displacements was presented and illustrated by means of one-dimensional examples by Rodrı́guez-Ferran
et al. (2005).
In this work, we extend the applicability of this alternative formulation to a twodimensional setting. First, we propose new boundary conditions which allow to regularise two-dimensional boundary problems. In addition, several examples are presented in order to validate the approach in a two-dimensional setting.
The structure of the chapter is as follows. In Section 3.1 the gradient version
of the damage model based on non-local displacements is briefly reviewed. Special
emphasis is placed on the definition of the boundary conditions for the regularisation
equation in Section 3.2. The regularisation capabilities are illustrated by means of
some numerical examples in Section 3.3. Finally, the concluding remarks in Section
3.4 close this chapter.
21
22
Continuous model of failure based on non-local displacements
3.1
Gradient continuous non-local damage model
In the implicit gradient-enhanced continuum model based on non-local displacements,
two different displacements are used to formulate the model: (a) the standard or
local displacements u a and (b) the gradient-enriched displacement field u g , which is
the solution of a partial differential equation with u a being the source term. This
regularisation PDE is the diffusion-reaction equation
ug (x
x, t) − `2 ∇2ug (x
x, t) = ua (x
x, t)
(3.1)
where ` is a parameter added in the diffusion term with the dimension of length.
The model is summarised in Table 3.1.
Table 3.1: Damage model based on non-local displacements, gradient version.
Constitutive equation
Local strains
Non-local displacements
Non-local strains
Non-local sate variable
Damage evolution
σ (x
x, t) = (1 − D (x
x, t)) C : ε (x
x, t)
x, t) = ∇su a (x
x, t)
ε (x
2 2
x, t) − ` ∇ u g (x
x, t) = u a (x
x, t)
u g (x
s
x, t) = ∇ u g (x
x, t)
ε g (x
x, t) = Y (εεg (x
x, t))
Yg (x
x, t) = D(Yg )
D (x
(3.2a)
(3.2b)
(3.2c)
(3.2d)
(3.2e)
(3.2f)
As Table 3.1 reflects, in this proposal, u g is an auxiliary regularised displacement
field that drives the damage evolution.
3.2
Boundary conditions for the regularisation equation
Integral-type formulations that fix spurious mesh sensitivity present some important
drawbacks, particularly when dealing with non-locality in the neighbourhood of the
3.2 Boundary conditions for the regularisation equation
23
boundary (Peerlings et al., 1996). As an alternative to these non-local descriptions,
gradient-enriched formulations have been used in the literature. The advantage of
using a gradient-enriched description is that although they are non-local models, they
are local from a mathematical viewpoint, since non-local interaction is accounted for
by means of higher-order derivatives. Nevertheless, the main disadvantage of gradient
approaches arises from the requirement of additional boundary conditions: in order to
solve the regularisation equation, boundary conditions have to be prescribed, which
is still an open issue in these formulations.
In standard gradient non-local damage models, homogeneous Neumann boundary
conditions
∇Ye · n = 0
on ∂Ω
(3.3)
(where n denotes the outward unit normal to Ω and Ye the non-local state variable) are
typically prescribed, due to the difficulty of motivating Dirichlet boundary conditions,
that is prescribing Ye .
In the damage model based on non-local displacements, the boundary conditions
seem to be easier to interpret. In this approach, a natural option is to prescribe
Dirichlet boundary conditions (Rodrı́guez-Ferran et al., 2005):
ug = ua
on ∂Ω
(3.4)
These boundary conditions have a clear physical interpretation: local and nonlocal displacements coincide along all the domain boundary (that is, for both the
Dirichlet and Neumann boundaries of the mechanical problem). However, as discussed in Jirásek and Marfia (2006), this may have the negative effect of not allowing
displacement smoothing along the boundary, since u g and u a are imposed to be equal
on ∂Ω. Such effect, especially negative in problems where localisation starts in the
boundary, may lead to spurious narrowing of the damage zone, as shown in Jirásek and
Marfia (2006), Rodrı́guez-Ferran et al. (2009) and Tamayo-Mas and Rodrı́guez-Ferran
(2010).
24
Continuous model of failure based on non-local displacements
In order to solve this problem, non-homogeneous Neumann boundary conditions
were suggested by Jirásek and Marfia (2006):
ug · n = ∇u
ua · n
∇u
on ∂Ω
(3.5)
(where n is the outward unit normal to the boundary). Note that by means of
boundary conditions (3.5), displacement smoothing along the boundary is accepted.
Nevertheless, boundary conditions (3.5) do not ensure volume conservation. Indeed, assuming a constant density and using the Gauss’ theorem, volume preservation
means
Z
Z
ug − u a ) dΩ =
∇ · (u
0=
Ω
ug − u a ) · n dΓ
(u
(3.6)
∂Ω
which is satisfied if Eq. (3.4) is employed but it is not fulfilled in general with boundary
conditions (3.5).
As an alternative to Eq. (3.4) and Eq. (3.5), combined boundary conditions are
proposed here: Dirichlet boundary conditions are prescribed for the normal component of the displacement field whereas Neumann boundary conditions are imposed for
the tangential one. Both homogeneous, Eq. (3.7), and non-homogeneous Neumann
boundary conditions, Eq. (3.8), are considered:
ug · n
= ua · n
ug · t ) · n =
∇ (u
ug · n
=
)
on ∂Ω
(3.7)
0
ua · n
ug · t ) · n = ∇ (u
ua · t ) · n
∇ (u
)
on ∂Ω
(3.8)
where n and t are the directions normal and tangent to the boundary ∂Ω respectively.
By means of essential boundary conditions (3.7)1 and (3.8)1 , volume preservation
is ensured. In addition, either homogeneous or non-homogeneous Neumann boundary
conditions allow displacement smoothing along the boundary: Eq. (3.7)2 allows a free
3.2 Boundary conditions for the regularisation equation
25
relative slip while the relative slips of local and non-local displacements are prescribed
to coincide if condition (3.8)2 is employed, which is a more restrictive condition. Note
that combined boundary conditions coincide with Dirichlet boundary conditions (3.4)
in a one-dimensonal setting.
Interestingly, if the local field u a is linear, the boundary value problem consisting
of the differential equation (3.1) and boundary conditions (3.4), (3.5) or (3.8) admits
solution u g = u a . Nevertheless, this is not the solution of Eq. (3.1) if combined
equations with homogeneous Neumann boundary condition for the tangential component are considered. Therefore, by means of Eq. (3.7), the modification of local
displacement fields into non-local ones does alter linear fields and since reproducibility
of order 1 is not ensured, spurious damage caused by small rigid rotations may occur.
In order to illustrate the above discussion, the regularisation equation
x) − `2 ∇2 ug (x
x) = ua (x
x)
ug (x
(3.9)
defined on the unit square [0, 1] × [0, 1] and three different scalar source terms ua are
considered.
As a first test, ua is assumed to be linear, see Figure 3.1(a). The boundary value
problem consisting of the differential equation (3.9) with the four proposed boundary
conditions is solved. As seen in Figure 3.1, given a linear function ua , solutions
x) = ua (x
x) are admitted if Dirichlet, Neumann or combined with non-homogeneous
ug (x
Neumann boundary conditions are employed, thus guaranteeing reproducibility of
x) = ua (x
x) is not the solution of Eq. (3.9) if boundary
order 1. Nevertheless, ug (x
conditions (3.7) are prescribed on the boundary, see Figure 3.1(d). As observed, in
order to fulfill the imposed boundary conditions, a high gradient solution is obtained
near the corners of the domain, which may lead to spurious damage caused by small
rigid rotations.
As a second test, ua is considered to be the function shown in Figure 3.2(a).
Again, the problem is solved using the four proposed boundary conditions. As shown
in Figure 3.2(b), Dirichlet boundary conditions do not allow displacement smoothing
along the boundary. By means of Neumann or combined boundary conditions, this is
26
Continuous model of failure based on non-local displacements
10
10
5
5
0
1
0
1
1
0.5
1
0.5
0.5
0.5
0 0
0 0
(a)
(b)
10
10
10
5
5
5
0
1
0
1
0
1
1
1
0.5
0.5
0.5
0.5
1
0.5
0.5
0 0
0 0
0 0
(c)
(d)
(e)
x) = ua (x, y) = 1 +
Figure 3.1: Validation test with (a) a linear source term ua (x
x + 5y. Solutions obtained by means of (b) Dirichlet, (c) Neumann, (d) Combined
(homogeneous Neumann) and (e) Combined (non-homogeneous Neumann) boundary
conditions.
permitted, see Figures 3.2(c), 3.2(d) and 3.2(e).
As a third test, the source term shown in Figure 3.3(a) is analysed. Again, Dirichlet
boundary conditions (3.4) do not allow a relative slip along the boundary, see Figure
3.3(b). However, this is permitted if Neumann or combined boundary conditions
are employed. As seen in 3.3(d), by means of homogeneous Neumann boundary
conditions, a free relative slip is allowed and no oscillations along the boundary appear.
3.3 Numerical examples: validation of the model
1
1
0.5
0.5
0
1
1
27
0
1
0.5
1
0.5
0.5
0.5
0 0
0 0
(a)
(b)
1
1
1
0.5
0.5
0.5
0
0
0
1
1
1
1
0.5
0.5
1
0.5
0.5
1
0.5
0.5
0 0
0 0
0 0
(c)
(d)
(e)
Figure 3.2: Validation test with (a) a tent function source term. Solutions obtained
by means of (b) Dirichlet, (c) Neumann, (d) Combined (homogeneous Neumann) and
(e) Combined (non-homogeneous Neumann) boundary conditions.
3.3
Numerical examples: validation of the model
The goal of this section is to illustrate the regularisation capabilities of the damage
model based on non-local displacements with combined boundary conditions (3.7).
Two different examples are carried out. In Section 3.3.1, a square plate under mode
I loading conditions is analysed and in Section 3.3.2, a single-edge notched beam test
is considered.
28
Continuous model of failure based on non-local displacements
1
1
0.5
0.5
0
1
0
1
1
1
0.5
0.5
0.5
0.5
0 0
0 0
(a)
(b)
1
1
1
0.5
0.5
0.5
0
1
0
1
0
1
1
0.5
0.5
1
1
0.5
0.5
0.5
0.5
0 0
0 0
0 0
(c)
(d)
(e)
Figure 3.3: Validation test with (a) a tent function source term. Solutions obtained
by means of (b) Dirichlet, (c) Neumann, (d) Combined (homogeneous Neumann) and
(e) Combined (non-homogeneous Neumann) boundary conditions.
3.3.1
Square plate under mode I loading conditions
The first example concerns the simulation of a pure mode I problem. It deals with
the solution of a square plate in tension subjected to a prescribed displacement at the
top and bottom side and clamped at the right one, see Figure 3.4. In order to cause
localisation, a weakened region whose size depends on the finite element discretisation
is considered.
The test is carried out according to a simplified Mazars criterion, see Equation
(2.4), and a linear softening law, see Equation (2.9). The dimensionless geometric
and material parameters for this test are summarised in Table 3.2.
3.3 Numerical examples: validation of the model
29
Figure 3.4: Square plate under mode I loading conditions: problem statement.
Table 3.2: Square plate under mode I loading conditions: geometric and material
parameters.
Meaning
Length of the specimen
Length of weaker part
Width of weaker part
Young’s modulus
Idem of weaker part
Poisson’s coefficient
Damage threshold
Final strain
Characteristic length
Symbol
L
LW
hW
E
EW
ν
Y0
Yf
`
Value
10
1
1 finite element
20 000
18 000 (10% reduction in E)
0
10−4
−2
1.25
√ × 10
7 × 10−4
The regularisation properties of the model are assessed by carrying out the analysis
with four different meshes of 10 × 11, 20 × 21, 30 × 31 and 40 × 41 elements, see Figure
3.5.
The damage profiles with the deformed meshes and the force-displacement curves
30
Continuous model of failure based on non-local displacements
(a) Mesh 1: 10 × 11 elements.
(b) Mesh 2: 20 × 21 elements.
(c) Mesh 3: 30 × 31 elements.
(d) Mesh 4: 40 × 41 elements.
Figure 3.5: Square plate under mode I loading conditions. Four meshes with different
element density and different imperfection sizes are used.
are shown in Figure 3.6 and 3.7 respectively. As seen, the force-displacement curve
and the width of damage band do not depend on numerical parameters such as the
finite element mesh or the imperfection size needed to cause localisation.
3.3.2
Single-edge notched beam
As a second example, a single-edge notched beam (SENB) subjected to an antisymmetrical four-point loading is considered. The geometry, loads and supports,
shown in Figure 3.8, correspond to the test carried out by (Rodrı́guez-Ferran and
Huerta, 2000). A plane stress analysis is performed. The test is carried out with the
modified von Mises model Eq. (2.5) with exponential damage evolution Eq. (2.7) and
the material parameters of Table 3.3.
As a first test, a fixed characteristic length ` =
√
10 mm is chosen. The analysis is
3.3 Numerical examples: validation of the model
(a) Mesh 1: 10 × 11 elements.
(b) Mesh 2: 20 × 21 elements.
(c) Mesh 3: 30 × 31 elements.
(d) Mesh 4: 40 × 41 elements.
31
Figure 3.6: Square plate under mode I loading conditions. Fixed characteristic length
with various meshes and imperfection sizes: damage profiles with deformed meshes
(×100).
carried out with four different meshes, see Figure 3.9. The final damage profiles and
the force-displacement curves are shown in Figure 3.10 and 3.11 respectively. As expected, regularisation via non-local displacements with combined boundary conditions
Eq. (3.7) solves the pathological mesh dependence.
As a second test, a fixed characteristic length ` = 10 mm is chosen. The numerical
analysis is carried out again with the four meshes shown in Figure 3.9. The forcedisplacement curves and the final damage profiles are shown in Figure 3.12 and 3.13
respectively. As seen, neither the force-displacement curve nor the width of damage
profiles depend on numerical parameters such as the finite element size. Moreover, as
32
Continuous model of failure based on non-local displacements
18
10x11 mesh
20x21 mesh
30x31 mesh
40x41 mesh
16
14
Force
12
10
8
6
4
2
0
0
0.02
0.04
0.06
0.08
Displacement
0.1
Figure 3.7: Square plate under mode I loading conditions. Fixed characteristic length
with various meshes and imperfection sizes: force-displacement curves.
Figure 3.8: Single-edge notched beam: problem statement. All distances in mm.
expected, the characteristic length ` controls the ductility of the material.
Nevertheless, as shown in Figure 3.13, some spurious damage appears at the
boundary. In fact, two different phenomena are observed. On the one hand, some
spurious damage emerges due to the punctual loads. Since in this second test the
characteristic length of the non-local technique is greater than in the first test, more
diffusion is introduced into the model and the width of this spurious damage is in-
3.4 Concluding remarks
33
Table 3.3: SENB reference test: material parameters for concrete beam (modified
von Mises model with exponential damage evolution) and steel loading plates (elastic
model).
Meaning
Young’s modulus
Poisson’s coefficient
Compressive-to-tensile strength ratio
Damage threshold
Residual strength
Slope of the soft. branch at peak
Symbol
E
ν
k
Y0
A
B
Concrete
28 000 MPa
0.1
10
1.5 × 10−4
0.8
9 000
Steel
280 000 MPa
0.2
10
creased. On the other hand, some physically unrealistic damage is observed at the
lower right-hand corner of the specimen. Although in this part of the specimen the
beam behaves as a rigid body and no deformation is observed, non-local strains are
nonzero thus leading to positive values of the non-local state variable, Eq. (3.2e), that
are able to damage the specimen, Eq. (3.2f).
In order to solve this pathological behaviour, which is due to the boundary conditions for the regularisation equation, the same test with combined equations and
non-homogeneous Neumann boundary conditions for the tangential component is going to be studied.
3.4
Concluding remarks
The main results of the present chapter can be summarised as follows:
• A gradient-enriched continuous formulation is employed in order to describe
the evolution of failure processes, between the undamaged state and macroscopic crack initiation. In this approach, two different displacement fields are
used: (a) the standard or local displacements u a and (b) the gradient-enriched
displacement field u g , which drives the damage evolution. This non-local displacement field is defined as the solution of a diffusion differential equation.
34
Continuous model of failure based on non-local displacements
Hence, additional boundary conditions should be prescribed.
• Dirichlet boundary conditions have a clear physical meaning: the two displacement fields are imposed to coincide along all the domain boundary. Nevertheless,
in a multi-dimensional setting, this leads to spurious narrowing of the damage
zone at the boundary, where displacement smoothing is not permitted.
• Although non-homogeneous Neumann boundary conditions allow this displacement smoothing, they are not expected to regularise the problem neither, since
volume conservation is not ensured.
• In order to solve the regularisation equation, combined boundary conditions can
be used. By means of these conditions, local and non-local displacements coincide along the normal direction to the boundary thus ensuring volume conservation. Moreover, some displacement along the tangent direction to the boundary
is allowed.
• Two different types of combined boundary conditions may be prescribed: either homogeneous or non-homogeneous Neumann boundary condition can be
imposed for the tangential component. By means of the homogeneous Neumann boundary condition, a free relative slip is permitted. However, if a nonhomogeneous Neumann boundary condition is prescribed, the relative slips of
local and non-local displacements coincide.
• If the local displacement field is linear, the regularisation equation with Dirichlet,
Neumann and combined boundary conditions with non-homogeneous Neumann
does admit solution u g = u a , thus guaranteeing reproducibility of order 1. Such
requirement is not ensured if combined equations with homogeneous Neumann
bounary condition are employed. So, the transformation of local displacement
fields into non-local ones modify linear fields.
• A pure mode I problem is carried out to exhibit the regularisation capabilities
of this strategy by means of combined boundary conditions with homogeneous
3.4 Concluding remarks
35
Neumann. As expected, the numerical results do not suffer from pathological
mesh sensitivity and physically realistic force-displacement diagrams and damage profiles are obtained.
• A mixed-mode problem is also analysed using two different characteristic lengths.
The results are quite successful if little diffusion is added into the model. However, at later stages of the process, spurious damage growth appears when a
greater characteristic length is employed. Such effect is caused by the boundary
conditions for the regularisation equation. In order to avoid them, combined
equations with non-homogeneous Neumann will be analysed in our further research.
36
Continuous model of failure based on non-local displacements
(a) Mesh 1: 407 elements, 465 nodes.
(b) Mesh 2: 763 elements, 841 nodes.
(c) Mesh3: 2997 elements, 3154 nodes.
(d) Mesh 4: 4244 elements, 4436 nodes.
Figure 3.9: SENB reference test. Four meshes with different element density are used.
3.4 Concluding remarks
37
(a) Mesh 1.
(b) Mesh 2.
(c) Mesh 3.
(d) Mesh 4.
Figure 3.10: SENB reference test. Fixed characteristic length ` =
various meshes: final damage profiles.
√
10 mm with
38
Continuous model of failure based on non-local displacements
60
Mesh 1
Mesh2
Mesh3
Mesh4
50
Force (kN)
40
30
20
10
0
0
0.02
0.04
CMSD (mm)
0.06
0.08
Figure 3.11: SENB reference test. Fixed characteristic length ` =
various meshes: force-displacement curves.
70
Mesh 1
Mesh2
Mesh3
Mesh4
60
50
Force (kN)
√
10 mm with
40
30
20
10
0
0
0.05
CMSD (mm)
0.1
Figure 3.12: SENB reference test. Fixed characteristic length ` = 10 mm with various
meshes: force-displacement curves.
3.4 Concluding remarks
39
(a) Mesh 1.
(b) Mesh 2.
(c) Mesh 3.
(d) Mesh 4.
Figure 3.13: SENB reference test. Fixed characteristic length ` = 10 mm with various
meshes: final damage profiles.
Chapter 4
Continuous-discontinuous model
based on non-local displacements
As already seen, by means of a non-local continuum approach, the damage inception
and its propagation can be simulated properly and physically realistic results are
obtained. Nevertheless, for increasing levels of damage, a continuum non-local model
is not able to simulate the physical discontinuities that can arise. For increasing loads,
this may cause an unrealistic spread of damage (Comi et al., 2007).
In order to solve this bad performance of continuum non-local techniques, a new
model is presented here: once the damage parameter exceeds a threshold set a priori,
we propose to couple the implicit gradient-enhanced damage continuum model with
a discontinuous approach allowing local displacements u a to admit discontinuities. In
addition, and for consistency purposes, see the regularisation equation (3.1), non-local
displacements are also modelled as discontinuous fields.
The structure of the chapter is as follows. In Section 4.1, the introduction of
discontinuities in the implicit gradient-enhanced continuum model based on non-local
displacements is presented. First, the definition of the discontinuous problem fields is
carried out in Section 4.1.1. Then, the governing equations are described in Section
4.1.2 and the variational formulation is derived in Section 4.1.3. Finally, the finite
element discretisation is presented in Section 4.1.4. Special emphasis is placed on
41
42 Continuous-discontinuous model based on non-local displacements
some finite element issues such as the crack-introduction or the crack-orientation
criteria in Section 4.2. The regularisation capabilities of this continuous-discontinuous
approach are illustrated in Section 4.3 by means of two numerical examples. Finally,
the concluding remarks in Section 4.4 close this chapter.
4.1
Gradient continuous-discontinuous non-local damage model
4.1.1
Problem fields
Consider the continuum body Ω bounded by Γ. The boundary Γ is composed of Γu ,
Γt and Γd such that Γ = Γu ∪Γt ∪Γd , as shown in Figure 4.1. Prescribed displacements
are imposed on Γu , while prescribed tractions are imposed on Γt . The boundary Γd
consists of the boundary of the crack.
Figure 4.1: Notations for a body with a crack subjected to loads and imposed displacements.
Then, in the body Ω̄, by means of the X-FEM, the standard displacement field ua
can be decomposed as
x) + HΓd (x
x) u 2a (x
x)
x) = u 1a (x
u a (x
(4.1)
x) (i = 1, 2) are continuous fields and HΓd is the Heaviside function centred
where u ia (x
4.1 Gradient continuous-discontinuous non-local damage model
43
at the discontinuity Γd . In this work, the sign function
(
x) =
HΓd (x
1 if x ∈ Ω̄+
−1 if x ∈ Ω̄−
(4.2)
(also called modified or generalised Heaviside function) centred at the discontinuity
surface Γd is employed, because of its symmetry (Zi and Belytschko, 2003).
A similar decomposition holds for the gradient-enriched displacements u g
x) = u 1g (x
x) + HΓd (x
x) u 2g (x
x)
u g (x
(4.3)
x) (i = 1, 2) are continuous fields and HΓd is the Heaviside function defined
where u ig (x
in Eq. (4.2).
Therefore, in the gradient-enhanced continuous-discontinuous model based on nonlocal displacements, two different displacements are used to formulate the model: (a)
the standard displacements u a and (b) the gradient-enriched displacements u g . By
means of the X-FEM, both fields are approximated by the sum of a continuous and
a discontinuous displacement field. The continuous part corresponds to the displacement field without any crack, while is the discontinuous or the enriched displacement
field the additional displacement that models the discontinuities.
4.1.2
Governing equations
The equilibrium equations and boundary conditions for the body Ω̄ without body
forces can be summarised as
∇·σ = 0
in Ω
(4.4a)
σ · n = t̄
on Γt
(4.4b)
σ · m = t̄ d
on Γd
(4.4c)
u a = u ∗a
on Γu
(4.4d)
44 Continuous-discontinuous model based on non-local displacements
where σ is the Cauchy stress tensor, n is the outward unit normal to the body, m is
the inward unit normal to Ω+ on Γd , u∗a is a prescribed displacement, t̄ is the load
on the boundary and t̄ d is the load on the discontinuity surface. Note that equation
(4.4c) represents traction continuity at the discontinuity surface Γd .
The strong form of the mechanical problem is completed by the damage constitutive relation
σ (x
x, t) = [1 − D (x
x, t)] C : ε (x
x, t)
(4.5)
In the regularisation approach employed in this work, the second-order diffusion
partial differential equation
x, t) − `2 ∇2u g (x
x, t) = u a (x
x, t)
u g (x
in Ω \ Γd
(4.6)
is coupled with the mechanical equations, also in a discontinuous setting. Both for
the standard and the enhanced displacement fields, boundary conditions must be
prescribed. Similarly as done in Section 3.2, different kinds of boundary conditions
may be imposed. On the one hand, Dirichlet
u ig = u ia
on ∂Ω ∪ Γd
(4.7)
or Neumann boundary conditions
uig · n = ∇u
uia · n
∇u
on ∂Ω ∪ Γd
(4.8)
where i = 1, 2, can be employed. As an alternative, combined boundary conditions are proposed here. Again, as suggested in Section 3.2, both homogeneous and
non-homogeneous Neumann boundary conditions are considered for the tangential
component of the displacement field:
u ig
·n
=
∇ u ig · t · n =
u ia

·n 
0

on ∂Ω
u ig
·m
=
∇ u ig · t · m =
u ia

·m 
0

on Γd
(4.9)
4.1 Gradient continuous-discontinuous non-local damage model
45


=
u ia · n
u ig · n
on ∂Ω
uia · t ) · n 
∇ u ig · t · n = ∇ (u
(4.10)


u ig · m
=
u ia · m
on Γd

i
i
u
t
m
u
t
m
∇ g· ·
= ∇ (u a · ) ·
where i = 1, 2.
4.1.3
Variational formulation
In this section, the variational formulation is derived: both the governing equations
(4.4) and the regularisation equation (4.6) with boundary conditions 4.9 are cast in a
weak form.
The space of trial local displacements is defined by the function
x, t) = u 1a (x
x, t) + HΓd (x
x)u
u2a (x
x, t) ,
u a (x
u 1a , u 2a ∈ Uu ,
(4.11)
where
Uu =
1 2
u a , u a | u 1a , u 2a ∈ H 1 (Ω) and u a |Γu = u ∗a
(4.12)
with H 1 (Ω) a Sobolev space.
The equilibrium equation (4.4a) is multiplied by the weight function
ω (x
x, t) = ω 1 (x
x, t) + HΓd (x
x)ω
ω 2 (x
x, t) ,
ω 1 , ω 2 ∈ Wu ,0
(4.13)
with
Wu ,0 =
1 2
ω , ω | ω 1 , ω 2 ∈ H 1 (Ω) and ω 1|Γu = ω 2|Γu = 0 ,
(4.14)
and integrated over the domain Ω to obtain the weak equilibrium statement. After
46 Continuous-discontinuous model based on non-local displacements
standard manipulations, the following expressions are obtained:
Z
s
1
Z
ω 1 · t̄ dΓ
Γt
ZΩ
Z
Z
s 2
2
HΓd ∇ ω : σ dΩ + 2
ω · t̄ d dΓ =
HΓd ω 2 · t̄t dΓ
ω 1 ∈ H 1 (Ω) (4.15a)
∀ω
∇ ω : σ dΩ =
Ω
Γd
ω 2 ∈ H 1 (Ω) (4.15b)
∀ω
Γt
where at the discontinuity,
ua K
t̄t˙ d = TJu̇
(4.16)
ua K.
with T relating traction rate t̄t˙ d and displacement jump rate Ju̇
Similarly to local displacements, the space of trial non-local displacements u g is
defined by the function
x, t) = u 1g (x
x, t) + HΓd (x
x)u
u2g (x
x, t) ,
u g (x
u 1g , u 2g ∈ Uu ,
(4.17)
where Uu is defined in Eq. (4.12).
Eq. (4.6) can be cast in a variational form by multiplication with the vector
x, t) Eq. (4.13) and integration over the domain Ω. After standard
test function ω (x
manipulations, one obtains
Z
1
u 1g
HΓd u 2g
2
Z
1
u1g
∇u
u2g
HΓd ∇u
2
Z
ω :
u2gm dΓ
ω ·
+
dΩ + `
∇ω
+
dΩ − `
HΓd ω 1 · ∇u
Ω
Ω
Γd
Z
Z
ω 1 ∈ Wu ,00
−2`2 ω 1 ∇ δΓd m · u 2g dΩ =
ω 1 · u 1g + HΓd u 2g dΩ
∀ω
(4.18a)
Ω
Ω
Z
Z
Z
2
2
2
2
1
2
1
2
ug dΩ − `
u1gm dΓ
ω : HΓd ∇u
ug + ∇u
HΓd ω 2 · ∇u
ω · HΓd u g + u g dΩ + `
∇ω
Ω
Ω
Z
Z Γd
u1g − HΓd ∇u
u2g − HΓd ∇ δΓd m · u 2g dΩ =
+2`2 ω 2 · δΓd m ∇u
ω 2 · HΓd u 1a + u 2a dΩ
Ω
Ω
ω 2 ∈ Wu ,00
∀ω
where δΓd is the Dirac delta centred at the discontinuity surface Γd .
(4.18b)
4.1 Gradient continuous-discontinuous non-local damage model
4.1.4
47
Discretised and linearised weak governing equations
In FE analysis, using a Galerkin discretisation, Eq. (4.1) and (4.3) read, for nodes
whose support is crossed by Γd ,
x) = N(x
x)u1a + HΓd (x
x)N(x
x)u2a
ua (x
(4.19a)
x) = N(x
x)u1g + HΓd (x
x)N(x
x)u2g
ug (x
(4.19b)
where N is the matrix of standard finite element shape functions, u1a/g are the basic
nodal degrees of freedom and u2a/g are the enhanced ones. The discrete format of the
problem fields leads to the four discrete weak governing equations
fint,u1a = fext,u1a
(4.20a)
fint,u2a = fext,u2a
(4.20b)
(M + `2 D)u1g + (MHΓd + `2 DHΓd )u2g = Mu1a + MHΓd u2a
(4.20c)
(MHΓd + `2 DHΓd )u1g + (M + `2 D)u2g = MHΓd u1a + Mu2a
(4.20d)
where
Z
fint,u1a =
ZΩ
fext,u1a =
BT σ dΩ
(4.21a)
NT t̄ dΓ
(4.21b)
Γt
Z
T
Z
HΓd B σ dΩ + 2
fint,u2a =
Ω
Z
fext,u2a =
NT t̄ d dΓ
(4.21c)
Γd
HΓd NT t̄ dΓ
(4.21d)
NT N dΩ
(4.21e)
∇NT ∇N dΩ
(4.21f)
HΓd NT N dΩ
(4.21g)
HΓd ∇NT ∇N dΩ
(4.21h)
Γt
Z
M =
ZΩ
D =
ZΩ
MHΓd =
DHΓd =
ZΩ
Ω
48 Continuous-discontinuous model based on non-local displacements
with B the matrix of shape function derivatives.
Some remarks about the discretisation:
• Eq. (4.20a) is the standard non-linear system of equilibrium equations, while
Eq. (4.20b) takes into account the contribution of the crack.
• In Eq. (4.21c), the contribution of the crack is multiplied by a factor of two due
to the chosen definition of the Heaviside function, see Eq. (4.2).
• Matrices M and D are the mass and diffusivity matrices already obtained in
(Rodrı́guez-Ferran et al., 2005). They are both constant.
• Matrices MHΓd and DHΓd can be understood as enriched mass and diffusivity
matrices respectively, since the expression is the same as M and D except for
the Heaviside function.
• Note that the property HΓd HΓd = +1, which is derived from the definition of
the Heaviside function Eq. (4.2), is used.
In summary, the finite element discretisation results in
requil,u1a := fint,u1a − fext,u1a = 0
(4.22a)
requil,u2a := fint,u2a − fext,u2a = 0
(4.22b)
rregu,u1g := (M + `2 D)u1g + (MHΓd + `2 DHΓd )u2g − Mu1a − MHΓd u2a = 0 (4.22c)
rregu,u2g := (MHΓd + `2 DHΓd )u1g + (M + `2 D)u2g − MHΓd u1a − Mu2a = 0 (4.22d)
where fint,u1a , fext,u1a , fint,u2a , fext,u2a are defined in Eq. (4.21a - 4.21d).
The consistent tangent matrix is

K 1 1
 ua ,ua
 K 2 1
 ua ,ua

 Ku1 ,u1
g a

Ku2g ,u1a
Ku1a ,u2a Ku1a ,u1g Ku1a ,u2g


Ku2a ,u2a Ku2a ,u1g Ku2a ,u2g 


Ku1g ,u2a Ku1g ,u1g Ku1g ,u2g 

Ku2g ,u2a Ku2g ,u1g Ku2g ,u2g
(4.23)
4.1 Gradient continuous-discontinuous non-local damage model
49
with the matrices defined in Table 4.1, so the linearised weak form at iteration i within
a time step k reads

k,i−1
k,i−1
k,i−1
Kk,i−1
u1a ,u1a Ku1a ,u2a Ku1a ,u1g Ku1a ,u2g
 k,i−1
k,i−1
k,i−1
 K 2 1 Kk,i−1
 ua ,ua
u2a ,u2a Ku2a ,u1g Ku2a ,u2g

k,i−1
 Ku1 ,u1 Kk,i−1
1 1
u1g ,u2a Kug ,ug Ku1g ,u2g
g a

k,i−1
2 1
2 2
Kk,i−1
u2g ,u1a Ku2g ,u2a Kug ,ug Kug ,ug

δu1a k,i

  δu2 k,i

a

  δu1g k,i

δu2g k,i
−rk,i−1
equil,u1a

 
  −rk,i−1
 
equil,u2a
=
 
0
 
0








(4.24)
Table 4.1: Matrices belonging to the consistent tangent matrix.
Ku1a ,u1a :=
R
Ω
Ku1a ,u1g := −
Ku2a ,u1a :=
R
R
Ω
Ku2a ,u1g := −
BT CB dΩ
g
BT CεεD0 (Yg ) ∂Y
B dΩ
∂εεg
Ω
HΓd BCB dΩ
R
Ω
g
B dΩ
HΓd BT CεεD0 (Yg ) ∂Y
∂εεg
Ku1a ,u2a :=
R
Ω
Ku1a ,u2g := −
Ku1a ,u2a :=
R
Ω
R
Ω
Ku2a ,u2g := −
HΓd BT CB dΩ
g
B dΩ
HΓd BT CεεD0 (Yg ) ∂Y
∂εεg
BT CB dΩ + 2
R
Ω
R
Γd
NT TN dΓ
g
B dΩ
BT CεεD0 (Yg ) ∂Y
∂εεg
Ku1g ,u1a := −M
Ku1g ,u2a := −MHΓd
Ku1g ,u1g := M + `2 D
Ku1g ,u2g := MHΓd + `2 DHΓd
Ku2g ,u1a := −MHΓd
Ku2g ,u2a := −M
Ku2g ,u1g := MHΓd + `2 DHΓd
Ku2g ,u2g := M + `2 D
Note that the arrays N and B multiplying u1a , u2a , u1g and u2g are not the same
since only part of the degrees of freedom in the arrays u1g and u2g are activated.
Some remarks about the tangent matrix (4.23):
• Matrix Ku1a ,u1a is the secant tangent matrix already obtained in (Rodrı́guezFerran et al., 2005). Matrices Ku1a ,u2a and Ku2a ,u1a may be understood as enriched
secant tangent matrices, since the expression is the same, except for the Heaviside function.
50 Continuous-discontinuous model based on non-local displacements
• Matrices Ku1a ,u1g and Ku2a ,u2g are the local tangent matrices already obtained in
(Rodrı́guez-Ferran et al., 2005). As done before, matrices Ku1a ,u2g and Ku2a ,u1g can
be understood as enriched local tangent matrices.
• Although the mass and diffusivity matrices are constant, the enriched ones may
increase during the numerical simulation, since the crack is allowed to propagate
through the bulk. Nevertheless, MHΓd and DHΓd are considered to be constant
during a fixed time step.
• Thanks to the linear relation between the degrees of freedom u1a , u2a , u1g and u2g ,
rregu,u1g and rregu,u2g are zero.
• Note again that, during the manipulations in Table 4.1, the property HΓd HΓd =
+1 is used.
4.2
Finite element technology
4.2.1
Introducing a discontinuity
In combined strategies, the transition between the continuous and the discontinuous
approach takes place when a critical situation is achieved, whose definition depends on
the underlying continuous model. In a damaging continuum approach, for example,
we will say that a critical situation is achieved when the damage parameter at one
integration point exceeds a critical damage value set a priori.
Therefore, as soon D > Dcrit , a crack described by a cohesive law is initiated,
damage value is fixed to Dcrit and the bulk material unloads, see Figure 4.2. In order
to preserve the robustness of the Newton-Raphson method, this crack is introduced
as a straight segment at the end of a time step. This procedure is repeated in the
elements ahead of the crack tip until the crack-introduction criterion is no longer
satisfied.
To model a crack tip, the displacement jump at the discontinuity tip is set to
zero. In order to prevent crack opening and sliding at the current crack tip, only
4.2 Finite element technology
(a)
51
(b)
Figure 4.2: Constitutive law for the cracked bulk: (a) elastic unloading with D = Dcrit ;
(b) linear traction-separation law for the crack.
standard degrees of freedom for the nodes of the edge containing the crack tip are
considered, see Figure 4.3. As soon the discontinuity is extended in the next element,
nodes behind the crack tip are enriched.
Figure 4.3: Sketch of the cohesive tip region. A crack line (dashed line) in a mesh with
standard elements (white) and elements whose nodes are enriched (grey). Enriched
nodes with H = +1 and H = −1 are indicated by squares and circles respectively.
4.2.2
Orienting a discontinuity
In a regularised continuous model, the crack growth direction cannot be analytically
derived, (Simone et al., 2003). In this work, we propose that the crack propagates
according to the steepest descent direction of the damage profile, see Figure 4.4.
52 Continuous-discontinuous model based on non-local displacements
Figure 4.4: Determination of the propagation direction: the crack propagates according to the direction perpendicular to the isolines of the damage field.
4.3
Numerical examples: validation of the model
The regularisation capabilities of this new strategy are illustrated in this section by
means of two numerical examples. In Section 4.3.1, a uniaxial tension test is carried
out using a two-dimensional geometry, and in Section 4.3.2 the two-dimensional square
plate under mode I loading conditions analysed in Section 3.3.1 is retrieved.
4.3.1
2D uniaxial tension test
This first example, Tamayo-Mas and Rodrı́guez-Ferran (2009), deals with the solution
of a bar in tension subjected to imposed displacement at the free side and clamped at
the other one, see Figure 4.5. Since in the first steps of the failure process a continuum
damage model is used, the central tenth of the bar is weakened to cause localisation.
A non-local continuum damage model is employed in the first stages of the failure
process, according to a simplified Mazars criterion Eq. (2.4) and a linear softening
law Eq. (2.9), see Figure 4.6(a). When the damage parameter exceeds a threshold set
a priori called Dcrit , a discontinuity is introduced and the continuous-discontinuous
technique is used. In order to characterise the crack, a linear traction-separation law
with secant unloading is considered, see Figure 4.6(b).
4.3 Numerical examples: validation of the model
53
Figure 4.5: Uniaxial tension test: problem statement.
(a)
(b)
Figure 4.6: Evolution law for (a) the bar; (b) the crack.
The dimensionless geometric and material parameters for this test are summarised
in Table 4.2.
Table 4.2: Uniaxial tension test with a non-local damage model and a softening
behaviour of the cohesive crack: geometric and material parameters.
Meaning
Length of the bar
Width of the bar
Length of weaker part
Young’s modulus
Idem of weaker part
Damage threshold
Final strain
Critical damage
Crack stiffness
Symbol
L
A
LW
E
EW
Y0
Yf
Dcrit
T
Value
100
1
L/7
20 000
18 000
10−4
1.25 × 10−2
0.9
−20
54 Continuous-discontinuous model based on non-local displacements
The regularisation properties of the model are analysed by means of different tests.
√
As a first test, a fixed characteristic length ` = 5 is chosen. The analysis is carried
out with six different meshes. The force-displacement curves and the damage profiles
are shown in Figure 4.7. As desired, the responses for this test do not depend on
finite element sizes.
651 Elem.
357 Elem.
105 Elem.
63 Elem.
21 Elem.
7 Elem.
Force
1.5
1
0.5
0
0
1
651 Elem.
357 Elem.
105 Elem.
63 Elem.
21 Elem.
7 Elem.
0.8
Damage
2
0.6
0.4
0.2
0.02
0.04
0.06
Displacement
0.08
0.1
0
0
20
(a)
40
X
60
80
100
(b)
Figure 4.7: Uniaxial tension test. Fixed characteristic length ` =
meshes. (a) force-displacement curve; (b) damage profiles.
√
5 with various
As a second test, a fixed mesh of 105 elements is considered and four different
√ √ √ √
characteristic lengths are used, ` = 1, 2, 5, 10. The results are depicted in
Figure 4.8. The ductility in the force-displacement response and the width of the
final damage profile increase with the internal length scale.
Finally, as a third test, a fixed mesh of 105 elements and a fixed characteristic
√
length ` = 5 are chosen. Two different tests, in which the size of weakened region
differs, are analysed. Results are shown in Figure 4.9. As seen, there is no pathological
dependence on imperfection size.
In summary, this new model exhibits the desired regularisation capabilities.
4.3.2
Square plate under mode I loading conditions
As a second example, the square plate shown in Figure 3.4 is retrieved. In order to
simulate the first steps of the failure process, the particular damage model presented
4.3 Numerical examples: validation of the model
2
1
0.5
0
0
2
l =10
l2=5
l2=2
l2=1
0.8
Damage
Force
1
2
l =10
l2=5
l2=2
l2=1
1.5
55
0.6
0.4
0.2
0.02
0.04
0.06
Displacement
0.08
0
0
0.1
20
(a)
40
X
60
80
100
(b)
Figure 4.8: Uniaxial tension test. Fixed mesh with various characteristic lengths. (a)
force-displacement curve; (b) damage profiles.
1
LW=L/7
LW=L/21
Damage
0.8
0.6
0.4
0.2
0
0
(a)
20
40
X
60
80
100
(b)
Figure 4.9: Uniaxial tension test. Fixed mesh and characteristic length with various
imperfection sizes. (a) force-displacement curve; (b) damage profiles.
in Section 3.3.1 is considered, see Table 3.2. When the damage parameter exceeds the
threshold Dcrit , which is set to 0.95, a growing crack is introduced and the combined
strategy is employed. To characterise the crack, a linear traction-separation law with
secant unloading is considered. In this particular test, the normal crack stiffness is
set to T = −208.33.
The regularisation properties of the model are assessed by carrying out the analysis
with three different meshes, see Figures 3.5(b)-3.5(d). The damage profiles with the
56 Continuous-discontinuous model based on non-local displacements
deformed meshes and the force-displacement curves are shown in Figure 4.10 and
4.11 respectively. As seen, the force-displacement curve and the width of damage
band do not depend on numerical parameters such as the finite element mesh or the
imperfection size needed to cause localisation.
(a) Mesh 2: 20 × 21 elements.
(b) Mesh 3: 30 × 31 elements.
(c) Mesh 4: 40 × 41 elements.
Figure 4.10: Square plate under mode I loading conditions. Fixed characteristic length
with various meshes and imperfection sizes: damage profiles with deformed meshes
(×100).
As shown in Figure 4.10, there exist some uncracked regions with D > Dcrit . In
order to fissure them, two main issues must be first taken into account. On the one
hand, the ability to deal with multiple non-intersecting cracks should be admitted
into our finite element code. On the other hand, crack branching phenomenon should
be modelled and also introduced into our finite element program.
4.4 Concluding remarks
57
Force
15
10
5
0
0
20x21 Mesh
30x31 Mesh
40x41 Mesh
2
4
Displacement
6
−3
x 10
Figure 4.11: Square plate under mode I loading conditions. Fixed characteristic length
with various meshes and imperfection sizes: force-displacement curves.
In addition, in Figure 4.11, some oscillations in the force-displacement diagrams
are observed. This behaviour is due to the use of X-FEM with Heaviside enrichment
only, which gives a binary description of the crack tip element (cracked or not cracked),
(Menouillard and Belytschko, 2010). Different methods have been developed in order
to avoid these spurious waves thus additional research is required to couple them with
the proposed combined strategy.
4.4
Concluding remarks
The main results of the present chapter can be summarised as follows:
• A gradient-enriched continuous formulation is enhanced with a discontinuous
interpolation of the problem fields in order to describe the final stages of failure processes, where macroscopic cracks arise. In this continuous-discontinuous
technique, both standard displacements u a and gradient-enhanced displacement
field u g may admit discontinuities.
• The eXtended Finite Element Method (X-FEM) is used to incorporate dis-
58 Continuous-discontinuous model based on non-local displacements
placement discontinuities in the numerical model thus requiring extra degrees
of freedom. In this proposal, the crack tip is assumed to belong to an element
edge and only the Heaviside function is employed in order to describe crack
propagation.
• The main features of this new combined strategy are summarised here:
– A continuous non-local damage model based on non-local displacements
is used to describe the first stages of failure. Therefore, numerical results
that do not present mesh sensitivity are obtained.
– At the end of each time step, the strategy checks if the transition criterion
is fulfilled. In this proposal, the transition between the continuous and the
discontinuous approach takes place when the damage parameter exceeds
a critical value set a priori. Hence, if D > Dcrit , a crack described by a
cohesive law is introduced. By now, only linear laws with secant unloadings
are considered.
– Once the crack is introduced, the discontinuous setting coexists with the
continuous one. In fact, here the continuum is used for crack path tracking:
the crack propagates according to the direction dictated by the gradient of
the damage field.
• In order to implement this discontinuous strategy, four different matrices are
needed: the standard M and the enhanced MHΓd mass matrices and the standard D and the enhanced DHΓd diffusivity matrices. The standard M and
D matrices, already obtained in (Rodrı́guez-Ferran et al., 2005), are constant.
However, the enhanced matrices may change during the numerical simulation,
since the crack propagates through the continuous bulk.
• Two different examples have been implemented to validate the strategy. On the
one hand, a one-dimensional problem is carried out. The expected regularisation
capabilities of the continuous-discontinuous model are obtained. On the other
hand, a two-dimensional problem under mode I loading conditions is studied.
4.4 Concluding remarks
59
By means of this example, it may be seen that the proposed combined approach
does regularise softening. Nevertheless, some further research is required in order
to extend its applicability to problems involving crack branching. Apart from
that, some spurious oscillations are observed in the force-displacement curves
due to the use of X-FEM with step enrichment only, which should be avoided.
Chapter 5
Future work
The final objective of this thesis is the computational analysis of a whole failure
process. Hence, to achieve this goal, different steps have been considered, which are
shown in the tentative schedule presented in Figure 5.1. Moreover, each of the future
work topics are detailed in the following list.
Figure 5.1: Work schedule.
• Boundary conditions for the regularisation equation
As explained in Section 3.2, prescribing appropriate boundary conditions for
the regularisation equation is still an open issue in these formulations. In this
proposal, four different boundary conditions are analysed: (a) Dirichlet boundary conditions, which do not allow displacement smoothing along the boundary;
61
62
Future work
(b) non-homogeneous Neumann boundary conditions, which do not ensure volume preservation and (c) two different kinds of combined boundary conditions,
which allow some displacement along the tangent direction to the boundary and
ensure volume conservation. On the one hand, homogeneous Neumann boundary conditions for the tangential component have been employed, which allow
a free relative slip but lack the ability to ensure reproducibility of order 1. On
the other hand, by means of non-homogeneous Neumann boundary conditions
for the tangential component, the relative slips of local and non-local displacements are prescribed to coincide. Although this is a more restrictive condition,
reproducibility of order 1 is guaranteed, thus preventing from spurious damage
caused by small rigid rotations.
As suggested by the above discussion, further research in this direction is needed.
On the one hand, these four boundary conditions are being analysed by means
of the single-edge notched beam. On the other hand, new numerical tests such
as the Nooru-Mohamed test (Nooru-Mohamed, 1992) will be carried out in order
to validate the model.
• Multiple non-intersecting discontinuities
By now, problems involving one single crack propagating through the continuous
bulk are analysed. Additional research is required in order to deal with n nonintersecting discontinuities. Indeed, if a body Ω̄ is crossed by n non-intersecting
cracks, both the standard u a and the enhanced u g displacement fields can be
decomposed as
x) = u 0a (x
x) +
u a (x
x) = u 0g (x
x) +
u g (x
n
X
x) u ia (x
x)
Hi (x
i=1
n
X
x) u ig (x
x)
Hi (x
(5.1a)
(5.1b)
i=1
where u ia and u ig , ∀ i = 0÷n, are continuous functions on Ω̄ and Hi are Heaviside
63
functions centred at the discontinuity surface Γi :
(
x) =
Hi (x
1 if x ∈ Ω̄+
i
(5.2)
−1 if x ∈ Ω̄−
i
Hence, more complicated problems such as a four-point bending test with n
notches will be carried out.
• Crack branching
The example presented in 4.3.2 is not completely simulated. The main reason
is that the proposed strategy lacks the ability to involve crack branching. Two
main issues arise when dealing with crack branching. On the one hand, different
strategies to model the branched element can be considered. In (Linder and
Armero, 2009), three of them are studied, see Figure 5.2.
(a) Element deletion.
(b) Element with a single
discontinuity.
(c) Element with embedded
branching.
Figure 5.2: Representation of the branching.
Although crack branching may not occur at an angle of 90◦ , the approximation
shown in Figure 5.2(c) is the most convenient one and it will be adopted in our
further research. On the other hand, the finite element implementation should
be studied in detail. The eXtended Finite Element Method (X-FEM) has been
successfully used in problems involving non-intersecting cracks. However, in the
case of branched and intersecting discontinuities, modelling them as independent is not suitable and a hierarchy between them should be considered, (Daux
et al., 2000; Zlotnik and Dı́ez, 2009). Different numerical techniques have been
64
Future work
developed in order to deal with crack branching, which will be investigated in
the near future.
• Application to other models
By now, only damage models to simulate failure of quasi-brittle materials have
been used. The idea is to extend the applicability of the proposed formulation
to a more generalised setting. On the one hand, plasticity models are going to
be studied. On the other hand, ductile damage and failure modelling will be
analysed.
Chapter 6
Publications
(1) Tamayo-Mas, E. and A. Rodrı́guez-Ferran, Continuous-discontinuous models of
failure based on non-local displacements, Congreso de Métodos Numéricos en
Ingenierı́a 2009, Barcelona (Spain). June 29-July 2, 2009.
(2) Rodrı́guez-Ferran, A., E. Tamayo-Mas, T. Bennett and H. Askes, A ContinuousDiscontinuous Model for Softening and Cracking based on Non-Local Gradient
Elasticity, Complas X - X International Conference on Computational Plasticity
Fundamentals and Applications, Barcelona (Spain). September 2-4, 2009.
(3) Tamayo-Mas, E. and A. Rodrı́guez-Ferran, A continuous-discontinuous model for
softening and cracking based on non-local displacements, ECCM 2010 - IV European Conference on Computational Mechanics, Paris (France). May 16-21, 2010.
65
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