miguel patrício cmuc polytechnic institute of leiria school of technology and management
TRANSCRIPT
![Page 1: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/1.jpg)
Miguel Patrício
CMUC
Polytechnic Institute of Leiria
School of Technology and Management
![Page 2: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/2.jpg)
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.
![Page 3: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/3.jpg)
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
![Page 4: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/4.jpg)
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
Why is cracking of composites worthy of attention?
![Page 5: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/5.jpg)
Fracture of composites can be regarded at different lengthscales
LENGTHSCALES
10-10 10-6 10-3 10-1 102
Microscopic(atomistic)
Mesoscopic Macroscopic
![Page 6: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/6.jpg)
Fracture of composites can be regarded at different lengthscales
LENGTHSCALES
10-10 10-6 10-3 10-1 102
Microscopic(atomistic)
Mesoscopic Macroscopic
Continuum Mechanics
![Page 7: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/7.jpg)
Macroscopic Mesoscopic(matrix+inclusions)
plate with pre-existent crack Meso-structure; linear elastic components
Goal: determine crack path
![Page 8: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/8.jpg)
homogenisation
Mesoscopic Macroscopic
It is possible to replace the mesoscopic structure with a corresponding homogenised structure (averaging process)
![Page 9: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/9.jpg)
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
![Page 10: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/10.jpg)
How will a crack propagate on a homogeneous (and isotropic) medium?
Crack tip
In the vicinity of a crack tip, the tangential stress is given by:
x
y
![Page 11: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/11.jpg)
How will a crack propagate on a homogeneous (and isotropic) medium?
Crack tip
In the vicinity of a crack tip, the tangential stress is given by:
x
y
![Page 12: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/12.jpg)
How will a crack propagate on a homogeneous (and isotropic) medium?
Maximum circumferential tensile stress (local) criterion:
Crack tip
“Crack growth will occur if the circumferential stress intensity factor equals or exceeds a critical value, ie.,”
Direction of propagation:
“Crack growth occurs in the direction that maximises the circumferential stress intensity factor”
x
y
![Page 13: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/13.jpg)
An incremental approach may be set up
solve elasticity problem;
load the plate;
The starting point is a homogeneous plate with a pre-existent crack
![Page 14: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/14.jpg)
An incremental approach may be set up
solve elasticity problem;
load the plate;
The starting point is a homogeneous plate with a pre-existent crack
...thus determining:
![Page 15: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/15.jpg)
An incremental approach may be set up
solve elasticity problem;
check propagation criterion;
compute the direction of propagation;
increment crack (update geometry);
If criterion is met
load the plate;
The starting point is a homogeneous plate with a pre-existent crack
![Page 16: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/16.jpg)
Incremental approach to predict whether and how crack propagation may occur
The mesoscale effects are not fully taken into consideration
![Page 17: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/17.jpg)
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 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
![Page 18: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/18.jpg)
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
![Page 19: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/19.jpg)
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
![Page 20: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/20.jpg)
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 Computational issues
number of graphite nodules in model: 113number of LTF zones in model: 31
Models were solved using Abaqus/Explicit (numerical package) running on a Beowulf Cluster with 8 Pentium 4 PCs
![Page 21: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/21.jpg)
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”
![Page 22: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/22.jpg)
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”
![Page 23: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/23.jpg)
Elasticity problem
Propagation problem
How will a crack propagate on a material with a mesoscopic structure?
![Page 24: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/24.jpg)
Elasticity problem Propagation problem
- Cauchy’s equation of motion
- Kinematic equations
- Constitutive equations
+ boundary conditions
- On a homogeneous material, the crack will propagate if
- If it does propagate, it will do so in the direction that maximises the circumferential stress intensity factor
many inclusions implies
high computational costs
the crackInteracts withthe inclusions
![Page 25: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/25.jpg)
HomogenisableHomogenisable
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
Hybrid approach
![Page 26: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/26.jpg)
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
![Page 27: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/27.jpg)
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
![Page 28: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/28.jpg)
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
![Page 29: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/29.jpg)
Reference cell
The material behaviour is characterisedby a tensor defined over the referencecell
How does homogenisation work?
Assumptions:
![Page 30: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/30.jpg)
Then the solution of the heterogeneous problem
![Page 31: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/31.jpg)
Then the solution of the heterogeneous problem
converges to the solution of a homogeneous problem
weakly in
![Page 32: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/32.jpg)
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
![Page 33: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/33.jpg)
a) 25 inclusions,
periodic
b) 100 inclusions,
periodic
c) 25 inclusions,
randomd) 100 inclusions,
random
![Page 34: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/34.jpg)
Periodical distributionof inclusions
Random distributionof inclusions
Highly heterogeneous composite with randomly distributed circular inclusions, submetido
Homogenisation may be employed to approximate the solution of the elasticity problems
Error decreases with number of inclusions
Error increases
![Page 35: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/35.jpg)
M. Patrício: Highly heterogeneous composite with randomly distributed circular inclusions, submitted
Smaller error
![Page 36: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/36.jpg)
plate (dimension 1x1) pre-existing crack (length 0.01) layered (micro)structure
E1=1, ν1=0.1
E2=10, ν2=0.3
![Page 37: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/37.jpg)
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)
![Page 38: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/38.jpg)
An iterative method for the prediction of crack propagation on highly heterogeneous media; M. Patrício, M. Hochstenbach, submitted
![Page 39: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/39.jpg)
Solve the elasticity problem
Is the crack tip on the matrix?
Compute the direction of propagation
Is the crack close to an inclusion?
Does the propagation angle point outwards?
Increment to reach crack interface, using
maximum circumferential tensile
stress criterion
Increment using maximum
circumferential tensile stress criterion
Propagate crack along the interface wall
![Page 40: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management](https://reader036.vdocument.in/reader036/viewer/2022062516/56649dea5503460f94ae51d7/html5/thumbnails/40.jpg)
Reference
Approximation