##
**Multi-scale micromorphic theory for hierarchical materials.**
*(English)*
Zbl 1159.74314

Summary: For the design of materials, it is important to faithfully model macroscopic materials response together with mechanisms and interactions occurring at the microstructural scales. While brute-force modeling of all the details of the microstructure is too costly, many of the current homogenized continuum models suffer from their inability to capture the correct underlying deformation mechanisms-especially when localization and failure are concerned. To overcome this limitation, a multi-scale continuum theory is proposed so that kinematic variables representing the deformation at various scales are incorporated. The method of virtual power is then used to derive a system of coupled governing equations, each representing a particular scale and its interactions with the macro-scale. A constitutive relation is then introduced to preserve the underlying physics associated with each scale. The inelastic behavior is represented by multiple yield functions, each representing a particular scale of microstructure, but collectively coupled through the same set of internal variables. The theory is illustrated by two applications. First, a one-dimensional example of a three-scale material is presented. After the onset of softening, the model shows that the localization zone is distributed according to two distinct length scale determined by the model. Second, a two-scale continuum model is introduced for the failure of porous metals. By comparing the theory to a direct numerical simulation (DNS) of the microstructure for a specimen in tension, we show that the model capture the main physics, and at the same time, remains computationally affordable.

### Keywords:

multi-scale micromorphic theory; microstructures; plastic collapse; inhomogeneous material; finite elements
PDF
BibTeX
XML
Cite

\textit{F. Vernerey} et al., J. Mech. Phys. Solids 55, No. 12, 2603--2651 (2007; Zbl 1159.74314)

Full Text:
DOI

### References:

[1] | Aifantis, E.C., Strain gradient interpretation of size effects, Int. J. fract., 95, 299-314, (1999) |

[2] | Bazant, Z., Why continuum damage is nonlocal: micromechanical arguments, J. eng. mech., 117, 1070-1087, (1991) |

[3] | Bazant, Z.; Jirasek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. eng. mech., 128, 1119-1149, (2002) |

[4] | Bishop, J.F.; Hill, R., A theory of the plastic distortion of polycrystalline aggregate under combined stresses, Philos. mag., 42, 414-427, (1951) · Zbl 0042.22705 |

[5] | Chambon, R.; Caillerie, D.; Matsuchima, T., Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies, Int. J. solids struct., 38, 8503-8527, (2001) · Zbl 1047.74522 |

[6] | Cosserat, E., Cosserat, F., 1909. Theorie des corps deformables, Paris: A Hermann et Fils. · JFM 40.0862.02 |

[7] | Eringen, A.C., Balance laws of micromorphic mechanics, Int. J. eng. sci., 8, 819-828, (1970) · Zbl 0219.73108 |

[8] | Fleck, N.A.; Hutchinson, J.W., Strain gradient plasticity, Adv. appl. mech., 33, 295-361, (1997) · Zbl 0894.73031 |

[9] | Fleck, N.A.; Muler, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta metall. mater., 42, 475-487, (1994) |

[10] | Forest, S.; Rainer, S.; Aifantis, E.C., Strain gradient crystal plasticity: thermomechanical formulations and applications, J. mech. behav. mater., 13, 219-232, (2002) |

[11] | Gao, H.; Huang, Y.; Nix, W.D.; Hutchinson, J.W., Mechanism-based strain gradient plasticity – theory, J. mech. phys. solids, 47, 1239-1263, (1999) · Zbl 0982.74013 |

[12] | Germain, P., The method of virtual power in continuum mechanics. part 2: microstrucure, SIAM J. appl. math., 25, 556-575, (1973) · Zbl 0273.73061 |

[13] | Gosh, S.; Lee, K.; Raghavan, P., A multi-level computational model for multi-scale damage analysis in composite and porous materials, Int. J. solids struct., 38, 2335-2385, (2001) · Zbl 1015.74058 |

[14] | Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth: part 1; yield criteria and flow rules for porous ductile media, ASME J. eng. mater. technol., 99, 2-15, (1977) |

[15] | Hao, S.; Liu, W.K.; Moran, B.; Vernerey, F.; Olson, G.B., Multiple-scale constitutive model and computational framework for the design of ultra-high strength, high toughness steels, Comput. meth. appl. mech. eng., 193, 1865, (2004) · Zbl 1079.74504 |

[16] | Huang, Y.; Zhang, T.F.; Guo, T.F.; Hwang, K.C., Mixed mode near-tip fields for cracks in materials with strain gradient effects, J. mech. phys. solids, 45, 439-465, (1997), (Methods in Applied Mechanics and Engineering, 193, p. 1865) · Zbl 1049.74521 |

[17] | Hutchinson, J.W., Plasticity at the micron scale, Int. J. solids struct., 37, 225-238, (2000) · Zbl 1075.74022 |

[18] | Kadowaki, H., Liu, W.K., 2004. Bridging Multi-Scale Method for Localization Problems, vol. 193, pp. 3267-3302. · Zbl 1060.74504 |

[19] | Kadowaki, H.; Liu, W.K., A multiscale approach for the micropolar continuum model, Comput. modeling eng. sci., 7, 269-282, (2005) · Zbl 1107.74004 |

[20] | Liu, W.K.; Karpov, E.G.; Zhang, S.; Park, H.S., An introduction to computational nanomechanics and materials, Comput. methods appl. mech. eng., 193, 1529-1578, (2004) · Zbl 1079.74506 |

[21] | Liu, W.K., Hao, S., Vernerey, F.J., Kadowaki, H., Park, H., Qian, D., 2004b. Multi- scale analysis and design in heterogeneous system. Presented at VII International Conference on Computational Plasticity (COMPLAS VII). |

[22] | Liu, W.K.; Karpov, E.G.; Parks, H.S., Nano mechanics and materials, theory, multiscale methods and applications, (2006), Wiley New York |

[23] | McVeigh, C.; Vernerey, F.; Liu, W.K.; Brinson, C., Multiresolution analysis for material design, Comput. methods appl. mech. eng., 195, 5053-5076, (2006) · Zbl 1118.74040 |

[24] | Michel, J.C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computationnal approach, Comput. methods appl. mech. eng., 143, 109-172, (1999) · Zbl 0964.74054 |

[25] | Mindlin, R.D., Micro-structure in linear elasticity, Arch. ration. mech. anal., 16, 15-78, (1984) · Zbl 0119.40302 |

[26] | Muhlaus, H.B.; Vardoulakis, I., The thickness of shear band in granular materials, Geotechnique, 237, 271-283, (1987) |

[27] | Nemat-Nasser, S., Hori, M., 1999. Micromechanics: overall properties of heterogeneous materials. · Zbl 0924.73006 |

[28] | Socrate, S., 1995. Mechanics of microvoid nucleation and growth in High-strength metastable austenitic steels. Ph.D. thesis, MIT. |

[29] | Tvergaard, V.; Needleman, A., Nonlocal effects on localization in a void-sheet, Int. J. solids struct., 34, 2221-2238, (1997) · Zbl 0942.74620 |

[30] | Vernerey, F.J., 2006. Multi-scale continuum theory for microstructured materials. Ph.D. thesis, Northwestern University. |

[31] | Vernerey, F.J.; McVeigh, C.; Liu, W.K.; Moran, B.; Tewari, D.; Parks, D.; Olson, G., Computational modeling of shear dominated ductile failure of steel, JOM, J. miner., met. mater. soc., 45-51, (2006) |

[32] | Wagner, G.J., Liu, W.K., 2003. Coupling of Atomistic and Continuum Simulations using a Bridging Scale Decomposition, vol. 190, pp. 249-274. · Zbl 1169.74635 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.