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

 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 Microscopic (atomistic) MesoscopicMacroscopic

 Fracture of composites can be regarded at different lengthscales LENGTHSCALES Microscopic (atomistic) MesoscopicMacroscopic Continuum Mechanics

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

homogenisation  Mesoscopic  Macroscopic  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., ” 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?  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?  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

 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:

 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

 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

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  Macrostructure  Computational issues 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

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 (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”

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 (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 crack Interacts with the inclusions

Homogenisable 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 , 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 , 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 , 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 , 2008  Hybrid approach algorithm

Reference cell The material behaviour is characterised by a tensor defined over the reference cell  How does homogenisation work? Assumptions:

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 distribution of inclusions Random distribution of 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 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

Reference Approximation