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Cracking

of composites. Cracking. Fundação Calouste Gulbenkian 15 a 17 de Julho de 2010. Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management. Composite materials.

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Cracking

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  1. of composites Cracking Fundação Calouste Gulbenkian 15 a 17 de Julho de 2010 Miguel Patrício CMUC Polytechnic Institute of Leiria School of Technology and Management

  2. Composite materials • 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.

  3. Composite materials • 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

  4. Cracking • 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

  5. Lengthscales • Fracture of composites can be regarded at different lengthscales Microscopic (atomistic) Mesoscopic Macroscopic 10-10 10-6 10-3 10-1 102 LENGTHSCALES

  6. Lengthscales • Fracture of composites can be regarded at different lengthscales Microscopic (atomistic) Mesoscopic Macroscopic 10-10 10-6 10-3 10-1 102 Continuum Mechanics LENGTHSCALES

  7. Problem formulation • plate with pre-existent crack • Meso-structure; linear elastic components • Goal: determine • crack path • Macroscopic • Mesoscopic • (matrix+inclusions)

  8. Macroscopic modelling • It is possible to replace the mesoscopic structure with a corresponding homogenised structure (averaging process) homogenisation • Mesoscopic • Macroscopic

  9. Macroscopic modelling • 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

  10. Macroscopic modelling • How will a crack propagate on a homogeneous (and isotropic) medium? • In the vicinity of a crack tip, the tangential stress is given by: y x • Crack tip

  11. Macroscopic modelling • How will a crack propagate on a homogeneous (and isotropic) medium? • In the vicinity of a crack tip, the tangential stress is given by: y x • Crack tip

  12. Macroscopic modelling • How will a crack propagate on a homogeneous (and isotropic) medium? • Maximum circumferential tensile stress (local) criterion: y “Crack growth will occur if the circumferential stress intensity factor equals or exceeds a critical value, ie.,” x • Direction of propagation: “Crack growth occurs in the direction that maximises the circumferential stress intensity factor” • Crack tip

  13. Incremental approach (macroscopic) • 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;

  14. Incremental approach (macroscopic) • 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:

  15. Incremental approach (macroscopic) • 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);

  16. Macroscopic modelling • Incremental approach to predict whether and how crack propagation may occur • The mesoscale effects are not fully taken into consideration

  17. Mesoscale modelling example • 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

  18. Mesoscale modelling example • 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

  19. Mesoscale modelling example • 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

  20. Mesoscale modelling example 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

  21. Computational limitations • 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. 453-463(11), 2002

  22. Computational limitations • 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. 453-463(11), 2002

  23. Mesoscopic problem • How will a crack propagate on a material with a mesoscopic structure? • Elasticity problem • Propagation problem

  24. Mesoscopic problem • Elasticity problem • Propagation problem • Cauchy’s equation of motion • On a homogeneous material, the crack will propagate if • Kinematic equations • If it does propagate, it will do so in the direction that maximises the circumferential stress intensity factor • Constitutive equations + boundary conditions many inclusions implies high computational costs the crack Interacts with the inclusions

  25. Solving the elasticity problem • Hybrid approach Homogenisable 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. 89-118, 2008

  26. Solving the elasticity problem • 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

  27. Solving the elasticity problem • 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

  28. Solving the elasticity problem • 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

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

  30. Homogenisation Then the solution of the heterogeneous problem

  31. Homogenisation Then the solution of the heterogeneous problem converges to the solution of a homogeneous problem weakly in

  32. Homogenisation (example) • 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

  33. Homogenisation (example) b) 100 inclusions, periodic a) 25 inclusions, periodic c) 25 inclusions, random d) 100 inclusions, random

  34. Homogenisation (example) • 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

  35. Hybrid approach (example) Smaller error M. Patrício: Highly heterogeneous composite with randomly distributed circular inclusions, submitted

  36. Numerical example • plate (dimension 1x1) • pre-existing crack (length 0.01) • layered (micro)structure E1=1, ν1=0.1 E2=10, ν2=0.3

  37. Numerical example • 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)

  38. Different microstructure An iterative method for the prediction of crack propagation on highly heterogeneous media; M. Patrício, M. Hochstenbach, submitted

  39. Different microstructure

  40. Different microstructure Reference Approximation

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