10 Miguel Patricio
Transcript of 10 Miguel Patricio
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Miguel PatrícioCMUC
Polytechnic Institute of LeiriaSchool of Technology and Management
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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.
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Idea: by putting together the right ingredients, in the right way, a material with a better performance can be obtainedExamples of applications: Airplanes Spacecrafts Solar panels Racing car bodies Bicycle frames Fishing rods Storage tanks
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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?
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Fracture of composites can be regarded at different lengthscales
LENGTHSCALES
10-10 10-6 10-3 10-1 102
Microscopic(atomistic)
Mesoscopic Macroscopic
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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
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Macroscopic Mesoscopic(matrix+inclusions)
plate with pre-existent crack Meso-structure; linear elastic components
Goal: determine crack path
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homogenisation
Mesoscopic Macroscopic
It is possible to replace the mesoscopic structure with a corresponding homogenised structure (averaging process)
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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.,”
whereis the strain energy released in the formation of a crack of length ais the corresponding surface energy increase
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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
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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
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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
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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
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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:
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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
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Incremental approach to predict whether and how crack propagation may occur
The mesoscale effects are not fully taken into consideration
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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
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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
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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
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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
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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”
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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”
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Elasticity problem
Propagation problem
How will a crack propagate on a material with a mesoscopic structure?
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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
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HomogenisableHomogenisable
Schwarz(overlapping domain decomposition scheme)
Critical region where fracture occursPatrício, 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
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Homogenisable
Critical region where fracture occursPatrício, 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
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Homogenisable
Critical region where fracture occursPatrício, 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
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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
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Reference cell
The material behaviour is characterisedby a tensor defined over the referencecell
How does homogenisation work?
Assumptions:
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Then the solution of the heterogeneous problem
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Then the solution of the heterogeneous problem
converges to the solution of a homogeneous problem
weakly in
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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
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a) 25 inclusions, periodic
b) 100 inclusions, periodic
c) 25 inclusions, random d) 100 inclusions,
random
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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
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M. Patrício: Highly heterogeneous composite with randomly distributed circular inclusions, submitted
Smaller error
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plate (dimension 1x1) pre-existing crack (length 0.01) layered (micro)structure
E1=1, ν1=0.1 E2=10, ν2=0.3
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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)
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An iterative method for the prediction of crack propagation on highly heterogeneous media; M. Patrício, M. Hochstenbach, submitted
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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
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Reference
Approximation