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Quantification of structural and material failure mechanisms across different length scales: from instability to brittle-ductile transitions

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Abstract

Structures may fail due to a myriad of different causes. Often, distinction is made between structural and material failure, that means a structure can fail, while the material is still intact (this is the case in so-called stability loss), or the material fails, which, as consequence, may lead to structural failure. The material behavior may turn out difficult to be mathematically guessed at the macro-level. On the other hand, a lot may be known about the chemistry or the microstructure of the material of interest. Herein, we aim at categorizing different scenarios which in the end provoke structural failure, discussing various cases investigated during the last five years, at the Institute for Mechanics of Materials and Structures of Vienna University of Technology: A well-chosen eigenvalue problem shows considerable potential for categorizing stability loss. We then turn to complex composite materials with a hierarchical organization, where a single constituent dominates the overall quasi-brittle failure of the material, such as lignin in wood and wood products, or the cement–water reaction products (shortly called hydrates) in cement-based materials. The picture changes if the first inelastically behaving constituent is related to ductile load carrying, then the loads within the microstructure are re-distributed before the overall material fails: this turns out to be the case in bone. Finally, due to highly confined multiaxial stress states, the elastic portion of the overall energy invested into the material may become negligible—and then yield design analysis employed on material volumes gives an idea of the highly ductile behavior of complex confined materials, such as asphalt. What integrates all the reported cases is the high capacity of mature mathematical and mechanical formulations to reveal the intricate, yet decipherable nature of the (continuum) mechanics of materials and structures.

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References

  1. Acker, P.: Micromechanical analysis of creep and shrinkage mechanisms. In: Ulm, F.-J., Bažant, Z.P., Wittmann, F.H. (eds.) Creep, Shrinkage and Durability Mechanics of Concrete and other Quasi-brittle Materials, 6th International Conference CONCREEP@MIT, pp. 15–26. Elsevier, Amsterdam (2001)

  2. Aigner E., Lackner R., Pichler C.: Multiscale prediction of viscoelastic properties of asphalt concrete. J. Mater. Civ. Eng. (ASCE) 21, 771–780 (2009)

    Article  Google Scholar 

  3. Bader T.K., Hofstetter K., Hellmich Ch., Eberhardsteiner J.: Poromechanical scale transitions of failure stresses in wood: from the lignin to the spruce level. ZAMM 90, 750–767 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bader T.K., Hofstetter K., Hellmich Ch., Eberhardsteiner J.: The poroelastic role of water in cell walls of the hierarchical composite “softwood”. Acta Mech. 217, 75–100 (2011)

    Article  MATH  Google Scholar 

  5. Barthèlèmy J.F., Dormieux L.: A micromechanical approach to the strength criterion of Drucker-Prager materials reinforced by rigid inclusions. Int. J. Numer. Anal. Methods Geomech. 28, 565–582 (2004)

    Article  MATH  Google Scholar 

  6. Benveniste Y.: A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech. Mater. 6, 147–157 (1987)

    Article  Google Scholar 

  7. Benveniste Y.: Revisiting the generalized self-consistent scheme in composites: clarification of some aspects and a new formulation. J. Mech. Phys. Solids 56, 2984–3002 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berliner R., Popovici M., Herwig K.W., Berliner M., Jennings H.M., Thomas J.J.: Quasielastic neutron scattering study of the effect of water-to-cement ratio on the hydration kinetics of tricalcium silicate. Cement Concr. Res. 28, 231–243 (1998)

    Article  Google Scholar 

  9. Bernard O., Ulm F.-J., Germaine J.T.: Volume and deviator creep of calcium-leached cement-based materials. Cement Concr. Res. 33, 1127–1136 (2003)

    Article  Google Scholar 

  10. Bernard O., Ulm F.-J., Lemarchand E.: A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cement Concr. Res. 33, 1293–1309 (2003)

    Article  Google Scholar 

  11. Bhowmik R., Katti K.S., Katti D.R.: Mechanics of molecular collagen is influenced by hydroxyapatite in natural bone. J. Mater. Sci. 42, 8795–8803 (2007)

    Article  Google Scholar 

  12. Bhowmik R., Katti K.S., Katti D.R.: Mechanisms of load-deformation behavior of molecular collagen in hydroxyapatite-tropocollagen molecular system: steered molecular dynamics study. J. Eng. Mech. 135, 413–421 (2009)

    Article  Google Scholar 

  13. Bodig J., Jayne B.: Mechanics of Wood and Wood Composites. Van Nostrand Reinhold, New York, USA (2011)

    Google Scholar 

  14. Budiansky B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227 (1965)

    Article  Google Scholar 

  15. Buehler M.J.: Nature designs tough collagen: explaining the nanostructure of collagen fibrils. Proc. Natl. Acad. Sci. USA (PNAS) 103, 12285–12290 (2006)

    Article  Google Scholar 

  16. Chatterji S., Jeffery J.W.: Three-dimensional arrangement of hydration products in set cement paste. Nature 209, 1233–1234 (1966)

    Article  Google Scholar 

  17. Chatterji S., Jeffery J.W.: Further studies of the three-dimensional arrangement of the hydration products of portland cements. Indian Concr. J. 45, 167–172 (1971)

    Google Scholar 

  18. Christiansen D.L., Huang E.K., Silver F.H.: Assembly of type I collagen: fusion of fibril subunits and the influence of fibril diameter on mechanical properties. Matrix Biol. 19, 409–420 (2000)

    Article  Google Scholar 

  19. Ciach T.D., Gillott J.E., Swenson E.G., Sereda P.J.: Microstructure of hydrated portland cement pastes. Nature 227, 1045–1046 (1970)

    Article  Google Scholar 

  20. Ciach T.D., Gillott J.E., Swenson E.G., Sereda P.J.: Microstructure of calcium silicate hydrates. Cement Concr. Res. 1, 13–25 (1971)

    Article  Google Scholar 

  21. Cusack S., Miller A.: Determination of the elastic constants of collagen by Brillouin light scattering. J. Mol. Biol. 135, 39–51 (1979)

    Article  Google Scholar 

  22. Crawley E.O., Evans W.D., Owen G.M.: A theoretical analysis of the accuracy of single-energy CT bone measurements. Phys. Med. Biol. 33, 1113–1127 (1988)

    Article  Google Scholar 

  23. Dormieux L., Kondo D., Ulm F.-J.: Microporomechanics. Wiley, Chichester (2006)

    Book  MATH  Google Scholar 

  24. Dormieux L., Molinari A., Kondo D.: Micromechanical approach to the behavior of poroelastic materials. J. Mech. Phys. Solids 50, 2203–2231 (2002)

    Article  MATH  Google Scholar 

  25. Dvorak G.J.: Transformation field analysis of inelastic composite materials. Proc. R. Soc. Lond. Ser. A 437, 311–327 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Eberhardsteiner, J.: Mechanisches Verhalten von Fichtenholz: Experimentelle Bestimmung der biaxialen Festigkeitseigenschaften [in German: Mechanical Behavior of Spruce Wood: Experimental Determination of Biaxial Strength Properties]. Springer, Wien, New York (2002)

  27. Fengel D., Wegener G., Wood G.: Wood—Chemistry, Ultrastructure, Reactions. De Gruyter, Berlin (1984)

    Google Scholar 

  28. Fritsch A., Hellmich Ch.: ‘Universal’ microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: micromechanics-based prediction of anisotropic elasticity. J. Theor. Biol. 244, 597–620 (2007)

    Article  Google Scholar 

  29. Fritsch A., Hellmich Ch., Dormieux L.: Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength. J. Theor. Biol. 260, 230–252 (2009)

    Article  Google Scholar 

  30. Füssl J., Lackner R.: Multiscale fatigue model for bituminous mixtures. Int. J. Fatigue 33, 1435–1450 (2011)

    Article  Google Scholar 

  31. Gao X.F., Lo Y., Tam C.M., Chung C.Y.: Analysis of the infrared spectrum and microstructure of hardened cement paste. Cement Concr. Res. 29, 805–812 (1999)

    Article  Google Scholar 

  32. Gentleman E., Lay A.N., Dickerson D.A., Nauman E.A., Livesay G.A., Dee K.C.: Mechanical characterization of collagen fibers and scaffolds for tissue engineering. Biomaterials 24, 3805–3813 (2003)

    Article  Google Scholar 

  33. Gross, D., Seelig, Th.: Bruchmechanik: Mit Einer Einführung in die Mikromechanik. [in German: Fracture mechanics: with an introduction to micromechanics]. Springer, Berlin, Germany, 3rd Edition (2001)

  34. Hamed E., Lee Y., Jasiuk I.: Multiscale modeling of elastic properties of cortical bone. Acta Mech. 213, 131–154 (2010)

    Article  MATH  Google Scholar 

  35. Hellmich Ch., Barthélémy J.-F., Dormieux L.: Mineral-collagen interactions in elasticity of bone ultrastructure – a continuum micromechanics approach. Eur. J. Mech. A-Solids 23, 783–810 (2004)

    Article  MATH  Google Scholar 

  36. Hellmich Ch., Kober C., Erdmann B.: Micromechanics-based conversion of CT data into anisotropic elasticity tensors, applied to FE simulations of a mandible. Ann. Biomed. Eng. 36, 108–122 (2008)

    Article  Google Scholar 

  37. Hellmich Ch., Ulm F.-J.: A micromechanical model for the ultrastructural stiffness of mineralized tissues. J. Eng. Mech. (ASCE) 128, 898–908 (2002)

    Article  Google Scholar 

  38. Helnwein, P.: Zur initialen Abschätzbarkeit von Stabilitätsgrenzen auf nichtlinearen Last-Verschiebungspfaden elastischer Strukturen mittels der Methode der Finite Elemente [in German; On ab initio assessability of stability limits on nonlinear load-displacement paths of elastic structures by means of the finite element method]. PhD thesis, Vienna University of Technology (1997)

  39. Hepworth D., Vincent J.: Modelling the mechanical properties of xylem tissues from tobacco plants (Nicotiana tabacum ’samsun’) by considering the importance of molecular and micromechanics. Ann. Bot. 81, 761–770 (1998)

    Article  Google Scholar 

  40. Hershey A.V.: The elasticity of an isotropic aggregate of anisotropic cubic crystals. J. Appl. Mech. (ASME) 21, 226–240 (1954)

    Google Scholar 

  41. Hill R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222 (1965)

    Article  Google Scholar 

  42. Höfinger, G.: Sensitivity Analysis of the Initial Postbuckling Behavior of Elastic Structures. PhD thesis, Vienna University of Technology (2010)

  43. Hofstetter K., Hellmich Ch., Eberhardsteiner J.: Development and experimental validation of a continuum micromechanics model for the elasticity of wood. Eur. J. Mech. A/Solids 24, 1030–1053 (2005)

    Article  MATH  Google Scholar 

  44. Hofstetter K., Hellmich Ch., Eberhardsteiner J.: Micromechanical modeling of solid-type and plate-type deformation patterns within softwood materials. A review and an improved approach. Holzforschung 61, 343–351 (2007)

    Article  Google Scholar 

  45. Hofstetter K., Hellmich Ch., Eberhardsteiner J., Mang H.A.: Micromechanical estimates for elastic limit states in wood materials, revealing nanostructural failure mechanisms. Mech. Adv. Mater. Struct. 15, 474–484 (2008)

    Article  Google Scholar 

  46. Kaczmarczyk L., Pearce C.J., Bićanić N., de Souza Neto E.: Numerical multiscale solution strategy for fracturing heterogeneous materials. Comput. Methods Appl. Mech. Eng. 199, 1100–1113 (2010)

    Article  MATH  Google Scholar 

  47. Katz J.L., Ukraincik K.: On the anisotropic elastic properties of hydroxyapatite. J. Biomech. 4, 221–227 (1971)

    Article  Google Scholar 

  48. Kollmann, F.: Technologie des Holzes und der Holzwerkstoffe [in German: Technology of Wood and Wood Products], 2nd edn., vol. 1. Springer, Berlin (1982)

  49. Kreher W.: Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Solids 38, 115–128 (1990)

    Article  MATH  Google Scholar 

  50. Kröner E.: Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls [in German: Computation of the elastic constants of a polycrystal based on the constants of the single crystal]. Zeitschrift für Physik A Hadrons and Nuclei 151, 504–518 (1958)

    Google Scholar 

  51. Lackner R., Blab R., Jäger A., Spiegl M., Kappl K., Wistuba M., Gagliano B., Eberhardsteiner J.: Multiscale modeling as the basis for reliable predictions of the behavior of multi-composed materials. Progr. Eng. Comput. Technol. Saxe-Coburg Publ. 8, 153–187 (2004)

    Google Scholar 

  52. Lafarge. Internal Report, Lafarge Centre Technique Europe Central, Commissioned by the Institute for Strength of Materials, Vienna University of Technology, Austria (1997)

  53. Lyamin A.V., Sloan S.W.: Lower bound limit analysis using non-linear programming. Int. J. Numer. Methods Eng. 55, 573–611 (2002)

    Article  MATH  Google Scholar 

  54. Mackenzie-Helnwein P., Eberhardsteiner J., Mang H.: A multi-surface plasticity model for clear wood and its application to the finite element analysis of structural details. Comput. Mech. 31, 204–218 (2003)

    Article  MATH  Google Scholar 

  55. Makrodimopoulos, A., Martin, C.M.: Limit analysis using large-scale SOCP optimization. In: Proceedings of the 13th National Conference of UK Association for Computational Mechanics in Engineering, Sheffield, pp. 21–24 (2005)

  56. Makrodimopoulos A., Martin C.M.: Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. Int. J. Numer. Methods Eng. 66, 604–634 (2006)

    Article  MATH  Google Scholar 

  57. Mang H.A., Höfinger G.: Bifurcation buckling from a membrane stress state. Int. J. Numer. Methods Eng. 90, 867–881 (2012)

    Article  MATH  Google Scholar 

  58. Mori T., Tanaka K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21, 571–574 (1973)

    Article  Google Scholar 

  59. Pan H., Tao J., Wu T., Tang R.: Molecular simulation of water behaviors on crystal faces of hydroxyapatite. Front. Chem. China 2, 156–163 (2007)

    Article  Google Scholar 

  60. Pichler B., Hellmich Ch.: Estimation of influence tensors for eigenstressed multiphase elastic media with nonaligned inclusion phases of arbitrary ellipsoidal shape. J. Eng. Mech. 136, 1043–1053 (2010)

    Article  Google Scholar 

  61. Pichler B., Hellmich Ch.: Upscaling quasi-brittle strength of cement paste and mortar: a multi-scale engineering mechanics model. Cement Concr. Res. 41, 467–476 (2011)

    Article  Google Scholar 

  62. Pichler B., Hellmich Ch., Eberhardsteiner J.: Spherical and acicular representation of hydrates in a micromechanical model for cement paste—prediction of early-age elasticity and strength. Acta Mech. 203, 137–162 (2009)

    Article  MATH  Google Scholar 

  63. Pichler B., Scheiner S., Hellmich Ch.: From micron-sized needle-shaped hydrates to meter-sized shotcrete tunnel shells: micromechanical upscaling of stiffness and strength of hydrating shotcrete. Acta Geotech. 3, 273–294 (2008)

    Article  Google Scholar 

  64. Pichler Ch., Lackner R., Aigner E.: Generalized self-consistent scheme for upscaling of viscoelastic properties of highly-filled matrix-inclusion composites—application in the context of multiscale modeling of bituminous mixtures. Composites Part B: Engineering 43, 457–464 (2012)

    Article  Google Scholar 

  65. Pillar, N.: Determination of Early Age Properties of Fibre Reinforced Shotcrete to Predict the Cracking Behavior. PhD thesis, submitted at University of New South Wales, Private Communication, Sydney, Australia (2002)

  66. Popper K.: Logik der Forschung [in German: Logic of Scientific Discovery]. Springer Wien, Austria (1934)

    Google Scholar 

  67. Powers T.C., Brownyard T.L.: Studies of the physical properties of hardened portland cement paste. Res. Lab. Portland Cement Assoc. Bull. 22, 101–992 (1948)

    Google Scholar 

  68. Richardson I.G.: Tobermorite/jennite- and tobermorite/calcium hydroxide-based models for the structure of C-S-H: applicability to hardened pastes of tricalcium silicate, β-dicalcium silicate, Portland cement, and blends of Portland cement with blast-furnace slag, metakaolin, or silica fume. Cement Concr. Res. 34, 1733–1777 (2004)

    Article  Google Scholar 

  69. Richartz W.: Über die Gefüge- und Festigkeitsentwicklung des Zementsteins [in German: Development of texture and strength in hardened cement paste]. Beton 19, 203–206 (1969)

    Google Scholar 

  70. Scheiner S., Sinibaldi R., Pichler B., Komlev V., Renghini C., Vitale-Brovarone C., Rustichelli F., Hellmich Ch.: Micromechanics of bone tissue-engineering scaffolds, based on resolution error-cleared computer tomography. Biomaterials 30, 2411–2419 (2009)

    Article  Google Scholar 

  71. Scrivener K.L.: Backscattered electron imaging of cementitious microstructures: understanding and quantification. Cement Concr. Compos. 26, 935–945 (2004)

    Article  Google Scholar 

  72. Steinböck A., Jia X., Höfinger G., Rubin H., Mang H.A.: Remarkable postbuckling paths analyzed by means of the consistently linearized eigenproblem. Int. J. Numer. Methods Eng. 76, 156–182 (2008)

    Article  MATH  Google Scholar 

  73. Steinböck A., Mang H.A.: Are linear prebuckling paths and linear stability problems mutually conditional?. Comput. Mech. 42, 441–445 (2008)

    Article  MATH  Google Scholar 

  74. Stroud A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs, NJ (1971)

    MATH  Google Scholar 

  75. Taplin J.M.: A method of following the hydration reaction in Portland cement paste. Aust. J. Appl. Sci. 10, 329–345 (1959)

    Google Scholar 

  76. Taylor R., Richardson I.G., Brydson R.M.D.: Nature of C-S-H in 20 year old neat ordinary Portland cement and 10 % Portland cement–90 % ground granulated blast furnace slag pastes. Adv. Appl. Ceram. 106, 294–301 (2007)

    Article  Google Scholar 

  77. Tritthart J., Häußler F.: Pore solution analysis of cement pastes and nanostructural investigations of hydrated C3S. Cement Concr. Res. 33, 1063–1070 (2003)

    Article  Google Scholar 

  78. Ullah,S., Pichler, B., Scheiner, S., Hellmich, Ch.: Influence of shotcrete composition on load-level estimation in NATM-tunnel shells: Micromechanics-based sensitivity analyses. Int. J. Numer. Anal. Methods Geomech. 5. doi:10.1002/nag.1043, available online at www3.interscience.wiley.com

  79. Vuong J., Hellmich Ch.: Bone fibrillogenesis and mineralization: quantitative analysis and implications for tissue elasticity. J. Theor. Biol. 287, 115–130 (2011)

    Article  Google Scholar 

  80. Williamson R.B.: Solidification of portland cement. Progr. Mater. Sci. 15, 189–286 (1972)

    Article  Google Scholar 

  81. Zahn D., Hochrein O.: Computational study of interfaces between hydroxyapatite and water. Phys. Chem. Chem. Phys. 5, 4004–4007 (2003)

    Article  Google Scholar 

  82. Zahn D., Hochrein O., Kawska A., Brickmann J., Kniep R.: Towards an atomistic understanding of apatite-collagen biomaterials: linking molecular simulation studies of complex-, crystal- and composite-formation to experimental findings. J. Mater. Sci. 42, 8966–8973 (2007)

    Article  Google Scholar 

  83. Zaoui A.: Continuum micromechanics: survey. J. Eng. Mech. (ASCE) 128, 808–816 (2002)

    Article  Google Scholar 

  84. Zienkiewicz O.C., Taylor R.L.: The Finite Element Method, Vol. 2, 4th Edn. McGraw-Hill, London, England (1994)

    MATH  Google Scholar 

  85. Zimmermann T., Sell J., Eckstein D.: Rasterelektronenmikroskopische Untersuchungen an Zugbruchflächen von Fichtenholz [in German: Scanning electron microscope investigation of the fracture surface of spruce]. Holz als Roh- und Werkstoff 52, 223–229 (1994)

    Article  Google Scholar 

  86. Zhang C., Cui M., Wang J., Gao X.W., Sladek J., Sladek V.: 3D crack analysis in functionally graded materials. Eng. Fract. Mech. 78, 585–604 (2011)

    Article  Google Scholar 

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  1. Institute for Mechanics of Materials and Structures, Vienna University of Technology, Vienna, Austria

    H. A. Mang, B. Pichler, T. Bader, J. Füssl, X. Jia, A. Fritsch, J. Eberhardsteiner & Ch. Hellmich

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  1. H. A. Mang
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  2. B. Pichler
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  3. T. Bader
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  4. J. Füssl
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  7. J. Eberhardsteiner
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  8. Ch. Hellmich
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Correspondence to J. Eberhardsteiner.

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Mang, H.A., Pichler, B., Bader, T. et al. Quantification of structural and material failure mechanisms across different length scales: from instability to brittle-ductile transitions. Acta Mech 223, 1937–1957 (2012). https://doi.org/10.1007/s00707-012-0685-1

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