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.

 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

 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

 Fracture of composites can be regarded at different lengthscales Microscopic (atomistic) Mesoscopic Macroscopic 10 -10 10 -6 10 -3 10 -1

LENGTHSCALES

10 2

 Fracture of composites can be regarded at different lengthscales Microscopic (atomistic) Mesoscopic Macroscopic 10 -10 10 -6 10 -3 10 -1

Continuum Mechanics

LENGTHSCALES

10 2

  plate with pre-existent crack Meso-structure; linear elastic components  Goal: determine crack path  Macroscopic  Mesoscopic (matrix+inclusions)

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

homogenisation

 Mesoscopic  Macroscopic

 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

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

y  In the vicinity of a crack tip, the tangential stress is given by: x  Crack tip

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

y  In the vicinity of a crack tip, the tangential stress is given by: x  Crack tip

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

y x  Maximum circumferential tensile stress (local) criterion: “

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

”  Crack tip

  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;

  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:

  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);

 Incremental approach to predict whether and how crack propagation may occur  The mesoscale effects are not fully taken into consideration

 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

 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

 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

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

 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 substantially limited by the computer elements is 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

 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 substantially limited by the computer elements is 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

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

 Elasticity problem  Propagation problem

 Elasticity problem Cauchy’s equation of motion - Kinematic equations - Constitutive equations + boundary conditions

many inclusions implies high computational costs

 Propagation problem - 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

the crack Interacts with the inclusions

 Hybrid approach Schwarz Critical region where fracture occurs (overlapping domain decomposition scheme) 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 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 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 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

 How does homogenisation work?

Reference cell The material behaviour is characterised by a tensor defined over the reference cell 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 unit stress and lower boundaries with constant

a) 25 inclusions, periodic c) 25 inclusions, random b) 100 inclusions, periodic d) 100 inclusions, random

 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

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

  plate (dimension 1x1) pre-existing crack (length 0.01)  layered (micro)structure E 1 =1, ν 1 =0.1 E 2 =10, ν 2 =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 Compute the direction of propagation Is the crack tip on the matrix?

Increment to reach crack interface, using maximum circumferential tensile stress criterion Is the crack close to an inclusion?

Increment using maximum circumferential tensile stress criterion Does the propagation angle point outwards?

Propagate crack along the interface wall

Reference Approximation