Abstract
Statistical Volume Elements (SVEs) are employed to evaluate homogenized mesoscopic ductile failure response of a carbon nanofiber reinforced composite under uniaxial tensile and compressive loadings. In the mesoscale analysis, after virtual reconstruction of the material microstructure, 2D finite element models are generated for each SVE using a non-iterative meshing algorithm named CISAMR, which fully automates the modeling process. The ductile damage response of each SVE is then simulated to derive its homogenized stress-strain curve. Corresponding initiation, maximum, and failure turning points, together with local fiber volume fraction and homogenized bulk modulus, are defined as mesoscopic Quantities of Interest (QoIs). While compressive loadings have slightly higher strength and nearly twice higher strains at failure compared to similar tensile cases, both loadings yield similar coefficients of variance for most QoIs. Further, a stochastic bulk damage model is calibrated from mesoscopic responses, which takes bilinear and linear forms versus strain for tensile and compressive loadings, respectively. Finally, cross-correlations are made between different QoIs, showing lower strain-based QoIs and higher strengths correspond to either higher local fiber volume fractions or more fibers being aligned with the loading direction.
Similar content being viewed by others
References
Ostoja-Starzewski M (2002) Microstructural randomness versus representative volume element in thermomechanics. J Appl Mech Trans ASME 69(1):25–35
Kanit T, Forest S, Galliet Ia, Mounoury Va, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13):3647–79
Liu WK, Siad L, Tian R, Lee S, Lee D, Yin X, Chen W, Chan S, Olson GB, Lindgen LE, Horstemeyer MF, Chang YS, Choi JB, Kim YJ (2009) Complexity science of multiscale materials via stochastic computations. Int J Numer Methods Eng 80(6–7):932–978
Sakata S, Ashida F, Zako M (2008) Kriging-based approximate stochastic homogenization analysis for composite materials. Comput Methods Appl Mech Eng 197(21–24):1953–1964
Kaminski M, Kleiber M (2000) Numerical homogenization of n-component composites including stochastic interface defects. Int J Numer Methods Eng 47(5):1001–1027
Sakata S, Ashida F, Enya K (2012) A microscopic failure probability analysis of a unidirectional fiber reinforced composite material via a multiscale stochastic stress analysis for a microscopic random variation of an elastic property. Comput Mater Sci 62:35–46
Sakata S, Ashida F, Ohsumimoto K (2013) Stochastic homogenization analysis of a porous material with the perturbation method considering a microscopic geometrical random variation. Int J Mech Sci 77:145–154
Xu H, Greene MS, Deng H, Dikin D, Brinson C, Liu KW, Burkhart C, Papakonstantopoulos G, Poldneff M, Chen W (2013) Stochastic reassembly strategy for managing information complexity in heterogeneous materials analysis and design. J Mech Des 135(10):
Baxter SC, Graham LL (2000) Characterization of random composites using moving-window technique. J Eng Mech 126(4):389–397
Huyse L, Maes MA (2001) Random field modeling of elastic properties using homogenization. J Eng Mech 127(1):27–36
Segurado J, Lorca JL (2006) Computational micromechanics of composites: the effect of particle spatial distribution. Mech Mater 38(8–10):873–883
Clarke PL, Abedi R, Bahmani B, Acton KA, Baxter SC (2017) Effect of the spatial inhomogeneity of fracture strength on fracture pattern for quasi-brittle materials. In: Proceedings of ASME 2017 international mechanical engineering congress & exposition IMECE 2017
Acton KA, Baxter SC, Bahmani B, Clarke PL, Abedi R (2018) Voronoi tessellation based statistical volume element characterization for use in fracture modeling. Comput Methods Appl Mech Eng 336:135–155
Dimas LS, Giesa T, Buehler MJ (2014) Coupled continuum and discrete analysis of random heterogeneous materials: elasticity and fracture. J Mech Phys Solids 63(1):481–490
Abedi R, Haber RB, Clarke PL (2017) Effect of random defects on dynamic fracture in quasi-brittle materials. Int J Fract 208(1–2):241–268
Genet M, Couegnat G, Tomsia AP, Ritchie RO (2014) Scaling strength distributions in quasi-brittle materials from micro- to macro-scales: A computational approach to modeling nature-inspired structural ceramics. J Mech Phys Solids 68(1):93–106
Al-Ostaz A, Jasiuk I (1997) Crack initiation and propagation in materials with randomly distributed holes. Eng Fract Mech 58(5–6):395–420
Kozicki J, Tejchman J (2007) Effect of aggregate structure on fracture process in concrete using 2D lattice model. Arch Mech 59(4–5):365–84
Yin X, Chen W, To A, McVeigh C, Liu WK (2008) Statistical volume element method for predicting microstructure-constitutive property relations. Comput Methods Appl Mech Eng 197(43–44):3516–3529
Bazant ZP, Planas J (1997) Fracture and size effect in concrete and other quasibrittle materials, vol 16. CRC Press, Boca Raton
Greene MS, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods Appl Mech Eng 200(1–4):309–325
Dubey V, Mashhadian M, Abedi S, Noshadravan A (2019) Multiscale poromechanical modeling of shales incorporating microcracks. Rock Mech Rock Eng 52(12):5099–5121
Silberschmidt VV (2006) Effect of micro-randomness on macroscopic properties and fracture of laminates. J Mater Sci 41(20):6768–6776
Bheemreddy V, Chandrashekhara K, Dharani LR, Hilmas GE (2016) Computational study of micromechanical damage behavior in continuous fiber-reinforced ceramic composites. J Mater Sci 51:8610–8624
Parambil NK, Gururaja S (2016) Micromechanical damage analysis in laminated composites with randomly distributed fibers. J Compos Mater 50(21):2911–2924
Zhi J, Zhao L, Zhang J, Liu Z (2016) A numerical method for simulating the microscopic damage evolution in composites under uniaxial transverse tension. Appl Compos Mater 23(3):255–269
Yang M, Nagarajan A, Liang B, Soghrati S (2018) New algorithms for virtual reconstruction of heterogeneous microstructures. Comput Methods Appl Mech Eng 338:275–298
Yang M, Ji M, Taghipour E, Soghrati S (2018) Cross-linked fiberglass packs: microstructure reconstruction and finite element analysis of the micromechanical behavior. Comput Struct 209:182–196
Soghrati S, Nagarajan A, Liang B (2017) Conforming to interface structured adaptive mesh refinement: new technique for the automated modeling of materials with complex microstructures. Finite Elem Anal Des 125:24–40
Soghrati S, Xiao F, Nagarajan A (2017) A conforming to interface structured adaptive mesh refinement technique for modeling fracture problems. Comput Mech 59(4):667–684
Nagarajan A, Soghrati S (2018) Conforming to interface structured adaptive mesh refinement: 3D algorithm and implementation. Comput Mech 62(5):1213–1238
Rashid YR (1968) Ultimate strength analysis of pre-stressed concrete pressure vessels. Nucl Eng Design 7(4):334–344
De Borst R (1984) Application of advanced solution techniques to concrete cracking and non-associated plasticity. Numer Methods Nonlinear Probl 2:314–325
Bažant ZP (1986) Mechanics of distributed cracking. Appl Mech Rev 39(5):675–705
Bažant ZP, Lin F-B (1988) Nonlocal smeared cracking model for concrete fracture. J Struct Eng 114(11):2493–2510
Camanho PP, Bessa MA, Catalanotti G, Vogler M, Rolfes R (2013) Modeling the inelastic deformation and fracture of polymer composites-Part II: Smeared crack model. Mech Mater 59:36–49
Pensée V, Kondo D, Dormieux L (2002) Micromechanical analysis of anisotropic damage in brittle materials. J Eng Mech 128(8):889–897
Kafka V (2004) Concrete under complex loading: mesomechanical model of deformation and of cumulative damage. Eur J Mech A/Solids 23(1):63–75
Ožbolt J, Li Y, Kožar I (2001) Microplane model for concrete with relaxed kinematic constraint. Int J Solids Struct 38(16):2683–2711
Bažant ZP, Caner FC (2005) Microplane model M5 with kinematic and static constraints for concrete fracture and an elasticity I: theory. J Eng Mech 131(1):31–40
Jefferson AD, Bennett T (2007) Micro-mechanical damage and rough crack closure in cementitious composite materials. Int J Numer Anal Methods Geomech 31(2):133–146
Bostanabad R, Liang B, Gao J, Liu WK, Cao J, Zeng D, Su X, Xu H, Li Y, Chen W (2018) Uncertainty quantification in multiscale simulation of woven fiber composites. Comput Methods Appl Mech Eng 338:506–532
Hooputra H, Gese H, Dell H, Werner H (2004) A comprehensive failure model for crashworthiness simulation of aluminium extrusions. Int J Crashworth 9(5):449–464
Sadowski T, Golewski P, Kneć M (2014) Experimental investigation and numerical modelling of spot welding-adhesive joints response. Compos Struct 112:66–77
Chen Z, Tang H, Shao Y, Sun Q, Zhou G, Li Y, Xu H, Zeng D, Su X (2019) Failure of chopped carbon fiber sheet molding compound (SMC) composites under uniaxial tensile loading: Computational prediction and experimental analysis. Compos Part A Appl Sci Manuf 118:117–130
Wu L, Nguyen V, Adam L, Noels L (2019) An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites. Comput Methods Appl Mech Eng 348:97–138
Bahmani B, Yang M, Nagarajan A, Clarke PL, Soghrati S, Abedi R (2019) Automated homogenization-based fracture analysis: effects of SVE size and boundary condition. Comput Methods Appl Mech Eng 345:701–727
Acton K, Sherod C, Bahmani B, Abedi R (2019) Effect of volume element geometry on convergence to a representative volume. ASCE-ASME J Risk Uncertain Eng Syst Part B Mech Eng 5(3):030907
Garrard JM, Abedi R (2020) Statistical volume element averaging scheme for fracture of quasi-brittle materials. Comput Geotech 117:103229
Hu A, Li X, Ajdari A, Jiang B, Burkhart C, Chen W, Brinson LC (2018) Computational analysis of particle reinforced viscoelastic polymer nanocomposites—statistical study of representative volume element. J Mech Phys Solids 114:55–74
McWilliams BA, Ramesh KT, Yen CF (2013) Probabilistic response of heterogeneous particle reinforced metal matrix composites with particle size dependent strengthening. Comput Mater Sci 79:15–24
Bessa MA, Bostanabad R, Liu Z, Hu A, Apley Daniel W, Brinson C, Chen W, Liu WK (2017) A framework for data-driven analysis of materials under uncertainty. Comput Methods Appl Mech Eng 320:633–667
Yang Z, Yabansu YC, Jha D, Liao W, Choudhary AN, Kalidindi SR, Agrawal A (2019) Establishing structure-property localization linkages for elastic deformation of three-dimensional high contrast composites using deep learning approaches. Acta Mater 166:335–345
Ma B, Ban X, Huang H, Chen Y, Liu W, Zhi Y (2018) Deep learning-based image segmentation for al-la alloy microscopic images. Symmetry 10(4):107
Cang R, Xu Y, Chen S, Liu Y, Jiao Y, Ren MY (2017) Microstructure representation and reconstruction of heterogeneous materials via deep belief network for computational material design. J Mech Des 139(7):
Mozaffar M, Bostanabad R, Chen W, Ehmann K, Cao J, Bessa MA (2019) Deep learning predicts path-dependent plasticity. Proc Natl Acad Sci USA 116(52):26414–26420
Shakiba M, Brandyberry DR, Zacek S, Geubelle PH (2019) Transverse failure of carbon fiber composites: analytical sensitivity to the distribution of fiber/matrix interface properties. Int J Numer Methods Eng 120(5):650–665
Zhang X, O’Brien DJ, Ghosh S (2019) Parametrically homogenized continuum damage mechanics (PHCDM) models for composites from micromechanical analysis. Comput Methods Appl Mech Eng 346:456–485
Sena MP, Ostoja-Starzewski M, Costa L (2013) Stiffness tensor random fields through upscaling of planar random materials. Probab Eng Mech 34:131–156
Garrard JM, Abedi R (2019) Statistical volume elements for the characterization of angle-dependent fracture strengths in anisotropic microcracked materials. J Risk Uncertain Eng Syst Part B (in press)
Zhu C, Zhu P, Liu Z (2019) Uncertainty analysis of mechanical properties of plain woven carbon fiber reinforced composite via stochastic constitutive modeling. Compos Struct 207:684–700
Carmeliet J, Hens H (1994) Probabilistic nonlocal damage model for continua with random field properties. J Eng Mech 120(10):2013–2027
Carmeliet J, de Borst R (1995) Stochastic approaches for damage evolution in standard and non-standard continua. Int J Solids Struct 32(8–9):1149–1160
Bahmani B, Abedi R, Clarke PL (2019) A stochastic bulk damage model based on Mohr–Coulomb failure criterion for dynamic rock fracture. Appl Sci 9(5):830
Malyarenko A, Ostoja-Starzewski M (2019) Towards stochastic continuum damage mechanics. Int J Solids Struct 184:202–210
Taylor LM, Chen EP, Kuszmaul JS (1986) Microcrack-induced damage accumulation in brittle rock under dynamic loading. Comput Methods Appl Mech Eng 55(3):301–320
Homand-Etienne F, Hoxha D, Shao JF (1998) A continuum damage constitutive law for brittle rocks. Comput Geotech 22(2):135–151
Shao JF, Rudnicki JW (2000) A microcrack-based continuous damage model for brittle geomaterials. Mech Mater 32(10):607–619
Lu YL, Elsworth D, Wang LG (2013) Microcrack-based coupled damage and flow modeling of fracturing evolution in permeable brittle rocks. Comput Geotech 49:226–44
Nguyen VP, Stroeven M, Sluys LJ (2011) Multiscale continuous and discontinuous modeling of heterogeneous materials: a review on recent developments. J Multisc Modell 3(04):229–270
Cai ZQ, Movva S, Chiou NR, Guerra D, Hioe Y, Castro JM, Lee LJ (2010) Effect of polyaniline surface modification of carbon nanofibers on cure kinetics of epoxy resin. J Appl Polym Sci 118(4):2328–2335
Lemaitre J, Chaboche JL, Maji AK (1993) Mechanics of solid materials. J Eng Mech 119(3):642–643
Piegl L, Tiller W (2012) The NURBS book. Springer, New York
Liang B, Nagarajan A, Ahmadian H, Soghrati S (2019) Analyzing effects of surface roughness, voids, and particle-matrix interfacial bonding on the failure response of a heterogeneous adhesive. Comput Methods Appl Mech Eng 346:410–439
Fiedler B, Hojo M, Ochiai S, Schulte K, Ando M (2001) Failure behavior of an epoxy matrix under different kinds of static loading. Compos Sci Technol 61(11):1615–1624
Ruzicka J, Spaniel M, Prantl A, Dzugan J, Kuzelka J, Moravec M (2012) Identification of ductile damage parameters in the ABAQUS. Bull Appl Mech 8:89–92
Hill R (1985) On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite strain. Mathematical proceedings of the Cambridge philosophical society, vol 98. Cambridge University Press, Cambridge, pp 579–590
Ahmadian H, Yang M, Nagarajan A, Soghrati S (2018) Effects of shape and misalignment of fibers on the failure response of carbon fiber reinforced polymers. Comput Mech 63(5):999–1017
Ren X, Chen JS, Li J, Slawson TR, Roth MJ (2011) Micro-cracks informed damage models for brittle solids. Int J Solids Struct 48(10):1560–1571
Jammalamadaka SR, Sengupta A (2001) Topics in circular statistics, vol 5. World Scientific, Singapore
Staber B, Guilleminot J (2017) Stochastic modeling and generation of random fields of elasticity tensors: a unified information-theoretic approach. Comptes Rendus Mécanique 345(6):399–416
Malyarenko A, Ostoja-Starzewski M (2018) Tensor-valued random fields for continuum physics. Cambridge University Press, Cambridge
Acknowledgements
Soghrati and Yang acknowledge the funding from Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-17-1-0350 and the U.S. National Science Foundation (NSF) under Grant No. 1608058, as well as the allocation of computing time from the Ohio Supercomputer Center (OSC). Abedi and Garrard acknowledge partial support for this work via the U.S. NSF, CMMI - Mechanics of Materials and Structures (MoMS) program Grant No. 1538332.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yang, M., Garrard, J., Abedi, R. et al. Effect of microstructural variations on the failure response of a nano-enhanced polymer: a homogenization-based statistical analysis. Comput Mech 67, 315–340 (2021). https://doi.org/10.1007/s00466-020-01934-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-020-01934-x