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What is Computational Micromechanics of Composites?

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  1. 1. NUMERICAL EXPERIMENTS IN MICROMECHANICS OF MATERIALS Leon Mishnaevsky Jr. Risø National Laboratory for Sustainable Energy, Technical University of Denmark
  2. 2. RISØ NATIONAL LABORATORY FOR SUSTAINABLE ENERGY Danish National Laboratory, 660 employee, and 8 departments. founded in 1956 by Niels Bohr, In 2007, Risø merged with the Technical University of Denmark
  3. 3. TECHNICAL UNIVERSITY OF DENMARK founded in 1829 by Hans Christian Ørsted (discoverer of electro-magnetism and aluminium) DTU ranks as the best Scandinavian (THES) and 5th best European university (Leiden ranking). 6000 students (2006), about 700 Ph D fellows and 580 full and associate professors
  4. 4. Group COMPOSITES at the Materials Research Department Development of wind Sino-Danish project UPWIND ”3D virtual testing of energy technologies in Nepal on the basis composites for wind of natural materials energy applications“ UpWind: Integrated Wind Turbine Design funded by EU, 40 participating Funded by Danish Agency institutions from 40 Sci., Technol. & Innovation, Funded by Royal Ministry countries; 2005-2011 2009-2010 of Denmark, ~700.000 EURO
  5. 5. TODAY’s TALK Can one optimize microstructures of composites? How to introduce microstructures into computational models? How to model damage in microstructural models? Computational models of several groups of composites Particle Reinforced Lightweight Composites Gradient Composites Interpenetrating Phase Composites Fiber Reinforced Composites Wood as a Hierarchical, Cellular Materials with Layered, Fibril Reinforced Cell Walls
  7. 7. GRADIENT, LAYERED AND SURFACE COMPOSITES FGM: Graded, smooth variation Coatings: one of the oldest Surface composites (Singh and of materials properties allow to technologies to improve the Fitz-Gerald): graded properties of increase the lifetime of materials reliability and lifetime of materials are achieved by transforming the under cyclic loading. and components. surface of the bulk material into truncated cone-like structures using Graded composites Example: Fatigue life of multiple pulse irradiation have 4..5 times higher stainless steels coated with technique+ deposition of the ZrNx increases by surface phase. service life. 400…1100%.
  8. 8. CLUSTERED/ „DOUBLE DISPERSION“ MICROSTRUCTURES Fine primary carbides lead to Coarse primary carbides are Double dispersion structure higher strength and lower wear- brittle and result in the higher ensures high wear-resistance resistance wear-resistance but lower and high strength strength Tool steels with “double dispersion structure” ensures 30% higher fracture toughness and 8 times higher lifetime, than a standard tool steel (s. Berns and colleagues, 1998)
  9. 9. NETLIKE MICROSTRUCTURES Inclusion networks can determine the crack path, and, thus, increase the fracture toughness of materials (Broeckmann, 1994, Gross-Weege et al, 1996)
  10. 10. HIERARCHICAL MATERIALS EXAMPLE: Synergy Ceramics Project by the Consortium of Japanese Universities. The idea is to create a new family of ceramic materials, by tailoring material properties using the “simultaneous control of different structural elements, such as shape and size, at plural scale levels”. Combining of aligned + anisotropic grains with the intragranular dispersion of nanoparticles in ceramics = gives high toughness and high strength (Kanzaki et al, 1999)
  11. 11. Properties of materials can be improved just by varying their microstructures. But how can we determine the optimal microstucture?
  12. 12. NUMERICAL EXPERIMENTS in the Mesomechanics of Materials Development Computational Testing Numerical of Microstructures Tools Acquiring Experimental Determination of Data Optimal Micro- structures and their Realization
  13. 13. CHALLENGES Numerical experiments require a large number of complex numerical models. How can they be generated? How to determine local properties of materials? How to introduce the complex microstructures into the numerical models?
  15. 15. HOMOGENIZATION Pierre Suquet (1987): In order to determine the constitutive equations for the averaged properties of a heterogeneous material One defines of a volume element, which is statistically representative for the whole microstructure. Localization (macro-micro transition): microscopic boundary conditions are determined on the basis of the macroscopic strain tensor. Homogenization (micro-macro transition): macroscopic properties of the equivalent homogeneous medium are determined on the basis of the analysis of the microscopic behavior of the RVE.
  16. 16. 1. Determination of RVE (Representative Volume Element). Unit cell models
  17. 17. RVE DESIGN: Classification of methods according to Professor Helmuth BÖHM (TU Vienna) Periodic Microfield Approaches (PMA) or Unit Cell (UC) Methods: assuming the periodic phase arrangement, one analyses a repeating unit cell in the microstructure Embedded Cell Approach: the materials is represented as a cut-out (unit cell) with a real microstructure, embedded into a region of the material with averaged properties “Windowing approach”: microstructure samples, chosen using “mesoscale test windows”, randomly placed in a heterogeneous material, are subject to homogeneous boundary conditions. By averaging the results for several “windows”, one can obtain bounds for the overall behavior of the material (Nemat-Nasser and Hori,1993) Modeling the full microstructure of a sample From: H.J. Böhm: Short Introduction to Basic Aspects of Continuum Micromechanics of Materials.Galway, Ireand; Juli 1998.
  19. 19. DESIGN OF MINIMAL UNIT CELLS Selection of UC, using symmetry analysis (after Li, 1999) Axisymmetric UC for particle reinforced composite:
  21. 21. AXISYMMETRIC UNIT CELLS WITH DAMAGE Debonding Crack in the particle Void (Mozhev and Kozhevnikova, 1997, Steglich and Brocks, 1997)
  23. 23. 2. Microstructure-based finite element model generation
  24. 24. MICROSTRUCTURE BASED MESH GENERATION Geometry- Voxel-based based mesh modelling: generation: ragged phase from digitized boundaries photos to FE mesh
  26. 26. VORONOI CELL FEM Voronoi-cell finite element method: a microstructure is divided into Voronoi polygons (a), which are then used as hybrid finite elements (b) (after Moorthy and Ghosh, 1998) Dirichlet tesselation Prescribed traction boundary Interelement boundary Prescribed displace-ment boundary
  27. 27. MULTIPHASE FINITE ELEMENTS: Multiphase finite elements: Interface may run across FE elements. phase boundary Integration points of one and the same element can be assigned to different phases. integration FE edges points Reconstruction of microstructures from serial sections, and generation of a microstructural model:
  28. 28. HIERARCHICAL MODEL: Example After Planckensteiner et al: the microstructure of a high speed steel with the carbide strings is modeled as a layered material at the mesolevel and as a statistically homogeneous two- phase material inside the strings
  29. 29. HOW TO INCLUDE DAMAGE IN CONTINUUM MICROMECHANICAL MODELS? Challenge: evolving physical discontinuities with infinitely small tips are incorporated into continuum mechanical, discretized problem.
  30. 30. CLASSIFICATION OF METHODS according to Professor Tony Ingraffea (Cornell) Non-geometrical representation: properties modification approaches: methods, based on the local reduction of element stiffness used to represent the crack path (“constitutive methods”, as computational cells, smeared crack, element elimination), and kinematic methods (xFEM, enriched elements), Geometrical representation: mesh modification approaches: constrained shape (i.e., if the crack path is prescribed by the faces of existing elements or by some theory-based assumptions) and arbitrary shape methods (meshfree, adaptive FEM/BEM, lattice methods, etc.).
  31. 31. MESH MODIFICATION APPROACHES Element elimination: The element is removed Nodal decoupling (often, followed by remeshing):
  32. 32. PROPERTY MODIFICATION APPROACHES Element weakening: Stiffness of an element is Smeared crack model: The reduced displacement jump is smeared out over some characteristic distance across the crack, which is correlated with the element size. The degradation of individual failure planes is described by the constitutive law.
  33. 33. UNIT CELLS PLACED ALONG THE CRACK PATH Computational cell model (after Xia and Shih): L Cell model of material (Broberg)
  34. 34. SPECIAL FINITE ELEMENT FORMULATIONS FE with special constitutive behavior: cohesive elements, described by traction-separation law. CE placed in the mesh in sites of potential damage initiation. FE with embedded discontinuities (Belytchko, Jirasek) Crack is simulated using the corresponding choice of the kinematic representation of localized fracture. The discontinuity, which crosses the element and divides it into two parts, is represented by additional degrees of freedom, corresponding to the normal and tangential components of the displacement jump. Generalized FEM (Babuška) combines the advantages of meshless methods and Nodal decoupling (often, followed the standard FEM. Taking into account that the nodal shape functions sum up to unity in the by remeshing): modeled area, they suggested to enrich the element shape functions by assumed local functions. eXtended FEM (XFEM): The displacement fields is presented as a sum of the regular displacement field (for the case without any discontinuities), and the enriched displacement field. Discontinuous enrichment functions are added to take into account the cracks and singular enrichment functions are added to account for the crack tips. Meshfree, other connectivity-free, adaptive methods. Galerkin (EFG) method for the discretization of structures, which are described by gradient-dependent damage models. Since the shape functions in the EFG method are formulated on the basis of the moving least squares principle, not on the basis of element connectivity, one can easily obtain higher order continuity shape functions. Askes et al. (2000) applied the element free
  35. 35. NUMERICAL TESTING OF MICROSTRUCTURES How to optimize microstructures of different groups of materials?
  36. 36. 1. Some software developed in our group Automatic 3D model generation
  37. 37. Automatic Generation of FE Models of 3D Microstructures 1. Free meshing + Geometry-based modelling for Particle Reinforced Composites Exact geometry. Best for testing artificial microstructures. 2. Voxel-based 3D reconstruction of microstructures Geometry is approximated by discrete voxels. 3. Fiber Reinforced Composites with Damageable elements and Interface Layer
  38. 38. 1st Code: FREE MESHING + 3D GEOMETRY- BASED MODEL Input data: (3, 3, 7) (6, 6, 2) volume content of particles, (2, 7, 2) .... type of particle arrangement (graded, cluster, random uniform, regular), particle shape (sphere, ellipsoid),average radius of particles, and standard deviation of radii. Output data: MSC/PATRAN Database statistical analysis of generated micro- structures
  39. 39. Automatic Generation and Meshing of Artificial 3D and 2D Microstructures Some Designed Microstructures: Localized Structures: graded Embedded Randomly arranged particle, with and clustered Cell FE constant and random sizes Model: Varied particle orientations: Varied gradient degrees: aligned, random, staggered
  40. 40. 2nd Code: VOXEL-BASED GENERATION OF 3D MICROSTRUCTURE MODELS Input data: [1,0,1, 0,0,0,0 1, 1 0,0,1, 0, 0, 1, 1, 1 voxel array, or ........................] statistical parameters of microstructure Output data: 3D microstructural FE model (MSC/PATRAN Database) The program carries out the percolation analysis of the microstructure.
  41. 41. 3rd Code: 3D MODEL OF FIBER REINFORCED COMPOSITES Input data: volume content and amount of fibers Output: MSC/PATRAN Database
  42. 42. ABAQUS Subroutine for Damage Simulation in Multiphase Materials ABAQUS Subroutine USDFLD : damage in element is simulated as local reduction of stiffness (element weakening), applies different damage criteria for different phases of the material: ⌧critical principal stress for brittle phases (particles), ⌧Lemaitre damage or critical strain failure condition (for matrix) stiffness of an element is reduced
  43. 43. 2. Experiments Before we simulate, we must know (a) local properties and (b) damage mechanisms
  44. 44. WHICH MICROMECHANISMS CONTROL DAMAGE AND FRACTURE OF MATERIALS? SEM in-situ Investigations of Micromechanisms of Deformation and Damage in AlSi cast Alloys and Tool Steels AlSi cast alloys 3-point bending specimen Tool steels load observed area How the micro- Primary carbides: before and after failure structure influen- ces the strength of the alloys? * Cracks are initiated by failure of Si particles caused by dislocation pile- ups. * Coalescence of cracks follows the shear bands. * Al cast alloys with globular microstructures have much higher failure strain than the alloys with lamellar microstructure. L. Mishnaevsky Jr et al., Eng. Fract. Mech. 63/ 4, 1999, pp. 395-411, Zeitschrift f. Metallkunde, 94, 2003, 6, pp. 676-681
  45. 45. HOW TO DETERMINE LOCAL PROPERTIES OF CONSTITUENTS AND PHASES? Hierarchical and inverse modelling Hierarchical (macro-micro) FE model of carbide failure in tool steels: micromodel includes the real microstructure macromodel reproduces the specimen Primary carbides: before & after failure applied load Type of the Cold High speed steel, High speed steel, steel work normal to the bands along the bands Failure stress of 1826 1604 2520 observed carbides area (MPa) L. Mishnaevsky Jr et al., Zeitschrift f. Metallkunde, 94, 2003, 6, pp. 676-681
  46. 46. Experimental-Numerical Methode: MAIN POINTS: SEM in-situ experiments: whereas the damage process is observed and recorded under SEM, the macroscopical F-U curve is recorded as well. Macro-micro simulation: the macroscopical model of the full specimen with a submodel (microstructural model) of the zone where the damage is observed. Inverse modeling: the strength and failure conditions of phases (local) is determined by comparing the micro-macro FE model with micro-macro observations in the experiments.
  47. 47. 3. Numerical testing of different microstructures Lightweight metal matrix (Al) composites reinforced by ceramics particles (SiC)
  48. 48. 3D Numerical Testing of Microstructures of Al/SiC Composites Distributions of Plastic Strains: on box Stress-strain curves & fraction of failed boundary particles vs. strain On particle/matrix interface Failure strain of composite: Failure strain of compo- sites increases in the In a vertical section following order: clustered < regular < random < gradient microstructure.
  49. 49. Mechanical Behavior of Polymer Composites (Polypropylene + Glass) Steps: Here: D0 is the instantaneous Homogenisation of the ε( t ) = g 0 (σe )D0 Se : σ( t ) + compliance, g0; g1; g2 and a are composite (nonlinear scalar functions of an equivalent t ⎛ t dt ' ⎞ viscoelastic matrix +elastic + g1 (σe ) ∫0 ΔD⎜ ∫τ ⎜ a (σ ) ⎟ ⎟ stress σ; ΔD(t) is a linear ⎝ e ⎠ viscoelastic creep compliance; ( )dτ particles), using tensors Se and Sc are 4th order affine formulation (tangent d g 2 (σe )Sc : σ(τ) tensors containing the elastic and linearisation of each phase) and Mori-Tanaka scheme dτ creep Poisson’s ratios. Implementation of 3D Shapery law into UMAT Comparison of the theoretical model with the numerical 3D model This work has been carried out together with Dr. M. Levesque and Prof. D. Baptiste (ENSAM, France). (see M. Levesque, et al., Composites Part A: Appl. Sci & Manuf, 35, 2004, 905-913)
  50. 50. 4. Fiber reinforced plolymer composites Competing damage mechanisms: fiber cracking, matrix fracture and interface damage L. Mishnaevsky Jr., P. Brøndsted, Composites Science and Technology, Vol. 69, No. 7-8, 2009, pp 1036-1044 Vol. 69, No3-4, 2009, pp. 477-484, Computational Materials Science, Vol. 44, No 4, 2009, pp 1351-1359
  51. 51. MODELLING OF DAMAGE IN FRC Fiber cracking: in Matrix cracking: in potential damageable ”interface layer” planes F M Fiber bridging: F I M Matrix crack growth from a fiber crack
  52. 52. MODELLING OF DAMAGE IN FRC Overloaded fibers near a failed fiber: Effect of variability of fiber properties: 400 350 300 Stress, MPa 250 200 Random (Weibull) 150 fiber strenths, viscoelastic matrix 100 Constant fiber 50 strength 0 0 0,005 0,01 0,015 0,02 0,025 0,03 Strain 3 Competing Damage Modes: Damage Evolution: Strong Matrix The interface crack is formed in the vicinity of a fiber crack Fiber cracking causes interface damage, and then leads to and the matrix crack is formed far away. Weak interface interface damage at neighbouring fiber delays matrix cracking!
  53. 53. 5. Functionally gradient composites What is the effect of microstructural gradient on strength and damage? L. Mishnaevsky Jr., Composites Sci. & Technology, 2006, Vol 66/11-12 pp 1873-1887
  54. 54. Numerical Testing of Generic Gradient Microstructures Design of Artificial Graded Von Mises Stress Distribution Gradient 3 microstructure Microstructures: Disp=1. Disp=5. Disp=15. Damage in particles and in matrix Varying the dispersion of the distribution, we can obtain highly gradient, as well as almost non- gradient particle distributions.
  55. 55. Effect of the Degree of Gradient on the Strength and Damage Evolution Fraction of failed particles vs. strain for different gradient degrees: Flow stress and stiffness of composites decrease, and the failure stress increases with increasing the gradient degree. Degree of homogeneity =1/Gradient degree
  56. 56. 6. Crack growth in tool steels Which arrangement of primary carbides ensures maximum toughness? L. Mishnaevsky Jr et al., Int. J. Fracture, Vol. 120, Nr. 4, 2003, pp. 581-600, Int. J. Fracture, 125: 33-50, 2004
  57. 57. NUMERICAL TESTING OF TOOL STEELS (1) Effect of Microstructure on the Fracture Toughness of Tool Steels FE Simulation of Crack Growth FE Simulation of Crack Growth in in the Real Microstructure Artificial Microstructures
  58. 58. NUMERICAL TESTING OF TOOL STEELS (2) Effect of Microstructure on the Fracture Toughness of Tool Steels 60 50 Band-like fine Net-like fine 40 Random fine 30 20 10 0 0 0,001 0,002 0,003 0,004 0,005 Displacement, mm Fracture energy is calcu- Fracture resistance of steels with layered & clustered lated as: G=Σ Pi ui/L, microstructures are higher than those with simple micro- where Pi - force and ui - dis- placement at increment i, L- structures. Net-like fine microstructure shows an exception to this rule. However, length of the microstructure such a mechanism of toughening (crack follows the carbide network) is unstable. area.
  59. 59. 7.Interpenetrating Phase Composites L. Mishnaevsky Jr., Materials Science & Engineering A, Vol. 407, No. 1-2, 2005, pp.11-23
  60. 60. OVERVIEW 3D cubic model by triangular prism unit cell model Daehn et al (1996) by Wegner and Gibson (2000). sphere interstiti 2-phase and 3-phase models by al matrix Feng et al. (2003, 2004) matricity model by Lessle, Dong and Schmauder
  61. 61. OUR APPROACH Unit cell model of interpenetrating phase composite Isotropic Gradient
  62. 62. EFFECT OF THE CONTIGUITY OF INTERPENETRATING PHASES Stress-strain curves & critical strain plotted versus vol. content of pcles No M percolation P & M percolation No P percolation Peak stress plotted versus maxim. cluster size Stiffness of composites increases almost linearly with increasing the maximum size of particle cluster up to the percolation threshold. A composite (ductile matrix + brittle inclusions) where the inclusions form a percolation cluster behaves as a brittle material.
  63. 63. GRADIENT INTERPENETRATING PHASE COMPOSITES Modeling of sharp/smooth graded interfaces Examples of the unit cells Stress-strain curves (examples): 2vc0 vc( y ) = 1 + e g − 2 gy / L Peak stress vs. sharpness of the interface Stiffness of graded composites increases, when the graded interface becomes smoother.
  64. 64. 8. Multiscale model of wood Wood as an hierarchical cellular material with layered cell walls and fibril reinforced wall sublayers H. Qing, L. Mishnaevsky Jr., Mechanics of Materials (in press), Comput. Matls Science (in press), Comput. Matls Science, Vol.44, 2, 2008, pp.363-370
  65. 65. HIERARCHICAL MODEL OF WOOD θ θ θ Halpin-Tsai model Multiscale model of wood: Mesolevel: the layered honeycomb like microstructure of cells is modelled as a 3D unit cell with layered walls. Submicrolevel: Each of the layers forming the cell walls was considered as an unidirectional, fibril reinforced composite.
  66. 66. HIERARCHICAL MODEL OF WOOD COMPUTATIONAL STUDIES SOME OBSERVATIONS Effect of Micrifibril Angle in S2 Layer The thickest and strong S2 on elastic properties sublayer is responsible for the shear strength, while strong and stiff “interphase” layers S1 and S3 are important to ensure the integrity of wood under XZ loading Microfibril angles in different sublayers of the cell wall control different properties of the wood Generally, different parameters of multiscale microstructure are Effect of cell shape on elastic responsible for different loading properties strengths.
  67. 67. 9. Wear of diamond grinding wheels
  68. 68. MESOMECHANICAL ANALYSIS OF WEAR OF DIAMOND GRINDING WHEELS Von Mises strain distribution on grinding wheel suface FE model of a cutout of Fraction of failed elements in wheel surface diamond grains versus force Mesomechanics approach is applicable to the analysis of the grinding and grinding wheel wear.
  69. 69. CONCLUSIONS Strength and damage resistance of materials can be improved by varying the microstructures of materials. The optimal microstructure of materials can be determined by using numerical experiments. A number of new numerical tools for the microstructural computational testing of materials have been developed and eployed for the numerical testing of microstructures: programs for the geometry-based and voxel-based generation of 3D microstructural model of composites, subroutines and programs for damage simulation, etc.
  70. 70. References: L. Mishnaevsky Jr, Computational Mesomechanics of Composites, Wiley, 2007, 290 pp. S. Schmauder, L. Mishnaevsky Jr, Micromechanics and Nanosimulation of Metals and Composites, Springer, 2008, 420 pp. L. Mishnaevsky Jr, Damage and Fracture of Heterogeneous Materials, Balkema, Rotterdam, 1998, 230 pp. Some papers are available on: http://risoe- About_risoe/ research_departments/ AFM/CV/lemi/lemi_cv/ news.aspx