Miguel Patrício
CMUC
Polytechnic Institute of Leiria
School of Technology and Management

Composites consist of two or more (chemically or
physically) different constituents that are bonded
together along interior material interfaces and do
not dissolve or blend into each other.

Idea: by putting together the right ingredients,
in the right way, a material with a better
performance can be obtained
Examples of applications:
 Airplanes
 Spacecrafts
 Solar panels
 Racing car bodies
 Bicycle frames
 Fishing rods
 Storage tanks
Why is cracking of composites worthy of
attention?




Even microscopic flaws
may cause seemingly safe
structures to fail
Replacing components of
engineering structures is
often too expensive and
may be unnecessary
It is important to predict
whether and in which
manner failure might
occur

Fracture of composites can be regarded at
different lengthscales
Microscopic
(atomistic)
10-10
Mesoscopic
10-6
Macroscopic
10-3
10-1
LENGTHSCALES
102

Fracture of composites can be regarded at
different lengthscales
Microscopic
(atomistic)
10-10
Mesoscopic
10-6
Macroscopic
10-3
10-1
Continuum Mechanics
LENGTHSCALES
102
plate with pre-existent crack

Meso-structure; linear elastic components
 Goal: determine
crack path


Macroscopic
Mesoscopic
(matrix+inclusions)

It is possible to replace the mesoscopic
structure with a corresponding homogenised
structure (averaging process)

homogenisation

Mesoscopic

Macroscopic

Will a crack propagate on a homogeneous
(and isotropic) medium?

Alan Griffith gave an answer for an infinite plate with a
centre through elliptic flaw:
“the crack will propagate if the strain energy release
rate G during crack growth is large enough to exceed
the rate of increase in surface energy R associated
with the formation of new crack surfaces, i.e.,”
where
is the strain energy released in the
formation of a crack of length a
is the corresponding surface
energy increase

How will a crack propagate on a
homogeneous (and isotropic) medium?
y

x

Crack tip
In the vicinity of a crack tip, the tangential
stress
is given by:

How will a crack propagate on a
homogeneous (and isotropic) medium?
y

x

Crack tip
In the vicinity of a crack tip, the tangential
stress
is given by:

How will a crack propagate on a
homogeneous (and isotropic) medium?
y

Maximum circumferential tensile stress (local)
criterion:
“Crack growth will occur if the circumferential stress
intensity factor equals or exceeds a critical value, ie.,”
x

Direction of propagation:
“Crack growth occurs in the direction that maximises
the circumferential stress intensity factor”

Crack tip


An incremental approach may be set up
The starting point is a homogeneous plate with a
pre-existent crack

load the plate;

solve elasticity problem;


An incremental approach may be set up
The starting point is a homogeneous plate with a
pre-existent crack

load the plate;

solve elasticity problem;
...thus determining:


An incremental approach may be set up
The starting point is a homogeneous plate with a
pre-existent crack

load the plate;

solve elasticity problem;

check propagation criterion;
If criterion is met


compute the direction of
propagation;
increment crack
(update geometry);


Incremental approach to predict whether and
how crack propagation may occur
The mesoscale effects are not fully taken into
consideration


In Basso et all (2010) the fracture toughness
of dual-phase austempered ductile iron was
analysed at the mesoscale, using finite
element modelling.
A typical model geometry consisted of a 2D
plate, containing graphite nodules and LTF
zones
Basso, A.; Martínez, R.; Cisilino, A. P.; Sikora, J.: Experimental and numerical assessment of fracture toughness of dual-phase austempered ductile iron,
Fatigue & Fracture of Engineering Materials & Structures, 33, pp. 1-11, 2010

Macrostructure

Mesostructure
Basso, A.; Martínez, R.; Cisilino, A. P.; Sikora, J.: Experimental and numerical assessment of fracture toughness of dual-phase austempered ductile iron,
Fatigue & Fracture of Engineering Materials & Structures, 33, pp. 1-11, 2010

Macrostructure

Results
Basso, A.; Martínez, R.; Cisilino, A. P.; Sikora, J.: Experimental and numerical assessment of fracture toughness of dual-phase austempered ductile iron,
Fatigue & Fracture of Engineering Materials & Structures, 33, pp. 1-11, 2010
number of graphite nodules in model: 113
number of LTF zones in model: 31
Models were solved using Abaqus/Explicit
(numerical package) running on a Beowulf
Cluster with 8 Pentium 4 PCs

Macrostructure

Computational issues
Basso, A.; Martínez, R.; Cisilino, A. P.; Sikora, J.: Experimental and numerical assessment of fracture toughness of dual-phase austempered ductile iron,
Fatigue & Fracture of Engineering Materials & Structures, 33, pp. 1-11, 2010

In Zhu et all (2002) a numerical simulation on
the shear fracture process of concrete was
performed:
“The mesoscopic elements in the specimen must be
relatively small enough to reflect the mesoscopic
mechanical properties of materials under the conditions
that the current computer is able to perform this analysis
because the number of mesoscopic elements is
substantially limited by the computer capacity”
Zhu W.C.; Tang C.A.: Numerical simulation on shear fracture process of concrete using mesoscopic mechanical model,
Construction and Building Materials, 16(8), pp. 453-463(11), 2002

In Zhu et all (2002) a numerical simulation on
the shear fracture process of concrete was
performed:
“The mesoscopic elements in the specimen must be
relatively small enough to reflect the mesoscopic
mechanical properties of materials under the conditions
that the current computer is able to perform this analysis
because the number of mesoscopic elements is
substantially limited by the computer capacity”
Zhu W.C.; Tang C.A.: Numerical simulation on shear fracture process of concrete using mesoscopic mechanical model,
Construction and Building Materials, 16(8), pp. 453-463(11), 2002

How will a crack propagate on a material with
a mesoscopic structure?

Elasticity problem

Propagation problem

Elasticity problem
- Cauchy’s equation of motion

Propagation problem
- On a homogeneous material, the crack
will propagate if
- Kinematic equations
- Constitutive equations
- If it does propagate, it will do so in
the direction that maximises the
circumferential stress intensity factor
+ boundary conditions
many inclusions
implies
high computational costs
the crack
Interacts with
the inclusions

Hybrid approach
Homogenisable
Schwarz
Critical region
where fracture
occurs
(overlapping
domain
decomposition
scheme)
Patrício, M.; Mattheij, R. M. M.; de With, G.: Solutions for periodically distributed materials with localized imperfections;
CMES Computer Modeling in Engineering and Sciences, 38(2), pp. 89-118, 2008

Hybrid approach
Homogenisable
Critical region
where fracture
occurs
Patrício, M.; Mattheij, R. M. M.; de With, G.: Solutions for periodically distributed materials with localized imperfections;
CMES Computer Modeling in Engineering and Sciences, 38(2), pp. 89-118, 2008

Hybrid approach
Homogenisable
Critical region
where fracture
occurs
Patrício, M.; Mattheij, R. M. M.; de With, G.: Solutions for periodically distributed materials with localized imperfections;
CMES Computer Modeling in Engineering and Sciences, 38(2), pp. 89-118, 2008

Hybrid approach algorithm
Patrício, M.; Mattheij, R. M. M.; de With, G.: Solutions for periodically distributed materials with localized imperfections;
CMES Computer Modeling in Engineering and Sciences, 38(2), pp. 89-118, 2008

How does homogenisation work?
Reference cell
The material behaviour is characterised
by a tensor defined over the reference
cell
Assumptions:
Then the solution of the heterogeneous problem
Then the solution of the heterogeneous problem
converges to the solution of a homogeneous problem
weakly in

Four different composites plates
(matrix+circular inclusions)

Linear elastic, homogeneous, isotropic constituents

Computational domain is [0, 1] x [0,1]


Material parameters:
matrix:
inclusions:
The plate is pulled along
its upper and lower
boundaries with constant
unit stress
a) 25 inclusions,
periodic
c) 25 inclusions,
random
b) 100 inclusions
periodic
d) 100 inclusions
random

Homogenisation may be employed to approximate
the solution of the elasticity problems
Periodical distribution
of inclusions
Error
increases
Error decreases with number of inclusions
Random distribution
of inclusions
Highly heterogeneous composite with randomly distributed circular inclusions, submetido
Smaller
error
M. Patrício: Highly heterogeneous composite with randomly distributed circular inclusions, submitted

plate (dimension 1x1)
pre-existing crack (length 0.01)

layered (micro)structure

E1=1, ν1=0.1
E2=10, ν2=0.3

plate (dimension 1x1)
pre-existing crack (length 0.01)

layered (micro)structure

Crack paths in composite materials; M. Patrício, R. M. M. Mattheij, Engineering Fracture Mechanics (2010)
An iterative method for the prediction of crack propagation on highly heterogeneous media; M. Patrício, M. Hochstenbach, submitted
Solve the elasticity
problem
Compute the direction
of propagation
Is the crack tip on
the matrix?
Does the
propagation angle
point outwards?
Is the crack close
to an inclusion?
Increment to reach
crack interface, using
maximum
circumferential tensile
stress criterion
Increment using
maximum
circumferential tensile
stress criterion
Propagate crack along
the interface wall
Reference
Approximation