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

Dec 25, 2015

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Page 1: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Then the solution of the heterogeneous problem

Page 31: Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Reference

Approximation