10 Miguel Patricio

Miguel PatrícioCMUC

Polytechnic Institute of LeiriaSchool 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 obtainedExamples of applications: Airplanes Spacecrafts Solar panels Racing car bodies Bicycle frames Fishing rods Storage tanks

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?

Fracture of composites can be regarded at different lengthscales

LENGTHSCALES

10-10 10-6 10-3 10-1 102

Microscopic(atomistic)

Mesoscopic Macroscopic

Fracture of composites can be regarded at different lengthscales

LENGTHSCALES

10-10 10-6 10-3 10-1 102

Microscopic(atomistic)


Continuum Mechanics

Macroscopic Mesoscopic(matrix+inclusions)

plate with pre-existent crack Meso-structure; linear elastic components

Goal: determine crack path

homogenisation


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

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

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

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

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


solve elasticity problem; load the plate;


...thus determining:


solve elasticity problem;

check propagation criterion;

compute the direction of propagation;

increment crack (update geometry);

If criterion is met

load the plate;


Incremental approach to predict whether and how crack propagation may occur

The mesoscale effects are not fully taken into consideration

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


Macrostructure Mesostructure


Macrostructure Results


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

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”

Elasticity problem

Propagation problem

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

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

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

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

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

Reference cell

The material behaviour is characterisedby a tensor defined over the referencecell

How does homogenisation work?

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

b) 100 inclusions, periodic

c) 25 inclusions, random d) 100 inclusions,

random

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

M. Patrício: Highly heterogeneous composite with randomly distributed circular inclusions, submitted

Smaller error

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

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

Reference

Approximation

10 Miguel Patricio

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