Cracking

1
Cracking
of composites
Fundação Calouste Gulbenkian 15 a 17 de Julho de
2010

• Miguel Patrício
• CMUC
• Polytechnic Institute of Leiria
• School of Technology and Management

2
Composite materials

• 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.

3
Composite materials

• 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

4
Cracking

• 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

5
Lengthscales

• Fracture of composites can be regarded at
  different lengthscales

Microscopic (atomistic)
Mesoscopic
Macroscopic
10-10
10-6
10-3
10-1
102
LENGTHSCALES
6
Lengthscales

• Fracture of composites can be regarded at
  different lengthscales

Microscopic (atomistic)
Mesoscopic
Macroscopic
10-10
10-6
10-3
10-1
102
Continuum Mechanics
LENGTHSCALES
7
Problem formulation

• plate with pre-existent crack

• Meso-structure linear elastic components

• Goal determine
• crack path

• Macroscopic

• Mesoscopic
• (matrixinclusions)

8
Macroscopic modelling

• It is possible to replace the mesoscopic
  structure with a corresponding homogenised
  structure (averaging process)

homogenisation

• Mesoscopic

• Macroscopic

9
Macroscopic modelling

• 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
10
Macroscopic modelling

• How will a crack propagate on a homogeneous (and
  isotropic) medium?

• In the vicinity of a crack tip, the tangential
  stress is given by

y
x

• Crack tip

11
Macroscopic modelling

• How will a crack propagate on a homogeneous (and
  isotropic) medium?

• In the vicinity of a crack tip, the tangential
  stress is given by

y
x

• Crack tip

12
Macroscopic modelling

• How will a crack propagate on a homogeneous (and
  isotropic) medium?

• Maximum circumferential tensile stress (local)
  criterion

y
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

13
Incremental approach (macroscopic)

• 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

14
Incremental approach (macroscopic)

• 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
15
Incremental approach (macroscopic)

• 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)

16
Macroscopic modelling

• Incremental approach to predict whether and how
  crack propagation may occur

• The mesoscale effects are not fully taken into
  consideration

17
Mesoscale modelling example

• 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
18
Mesoscale modelling example

• 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
19
Mesoscale modelling example

• 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
20
Mesoscale modelling example
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
21
Computational limitations

• 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
22
Computational limitations

• 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
23
Mesoscopic problem

• How will a crack propagate on a material with a
  mesoscopic structure?

• Elasticity problem

• Propagation problem

24
Mesoscopic problem

• Elasticity problem

• Propagation problem

• Cauchys equation of motion

• On a homogeneous material, the crack will
  propagate if

• Kinematic equations

• If it does propagate, it will do so in the
  direction that maximises the circumferential
  stress intensity factor

• Constitutive equations

boundary conditions
many inclusions implies high computational costs
the crack Interacts with the inclusions
25
Solving the elasticity problem

• Hybrid approach

Schwarz (overlapping domain
decomposition scheme)
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
26
Solving the elasticity problem

• Hybrid approach

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
27
Solving the elasticity problem

• Hybrid approach

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
28
Solving the elasticity problem

• 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
29
Homogenisation

• How does homogenisation work?

Reference cell The material behaviour is
characterised by a tensor defined over the
reference cell
Assumptions
30
Homogenisation
Then the solution of the heterogeneous
problem
31
Homogenisation
Then the solution of the heterogeneous
problem converges to the solution of a
homogeneous problem weakly in
32
Homogenisation (example)

• Four different composites plates
• (matrixcircular 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

33
Homogenisation (example)
b) 100 inclusions, periodic
a) 25 inclusions, periodic
c) 25 inclusions, random
d) 100 inclusions, random
34
Homogenisation (example)

• 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
35
Hybrid approach (example)
Smaller error
M. Patrício Highly heterogeneous composite with
randomly distributed circular inclusions,
submitted
36
Numerical example

• plate (dimension 1x1)

• pre-existing crack (length 0.01)

• layered (micro)structure

E11, ?10.1
E210, ?20.3
37
Numerical example

• 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)
38
Different microstructure
An iterative method for the prediction of crack
propagation on highly heterogeneous media M.
Patrício, M. Hochstenbach, submitted
39
Different microstructure
40
Different microstructure
Reference
Approximation