Elastic Behavior of 2D Grain Packing Modeled as Micromorphic Media Based on Granular Micromechanics
Publication: Journal of Engineering Mechanics
Volume 143, Issue 1
Abstract
Discrete simulation and continuum modeling are two competing methods for analyzing the behavior of complex material systems and granular assemblies. In this paper, a two-dimensional (2D) granular micromechanics model for micromorphic media is presented. The granular micromechanics method is enhanced by incorporating nonclassical terms in the kinematics. These include, in addition to the usual macroscale displacement gradient, the fluctuations in displacement gradient and also the second gradient of displacement. Microscopic intergranular force vectors and macroscopic stress tensors conjugate to the kinematic measures are defined, and the macroscopic strain energy density function is set equal to the volume average of grain-pair energies. As a result, continuum stiffness tensors are formulated on the basis of intergranular stiffness coefficients and fabric parameters defining the geometry of grains and their contacts. To demonstrate the applicability of the continuum method, a random granular assembly is analyzed with both the developed model and discrete modeling method. Results of the discrete analysis then are used to identify the macroscale (or continuum) and the microscale (or grain-pair) stiffness coefficients for a randomly generated assembly of disks. Effects of the three kinematic fields, macrodisplacement gradient, fluctuations in displacement gradient, and its second gradient, are compared. It is found that the derived grain-scale stiffness coefficients are not equal to the ones used in the discrete simulation. This implies that the intergranular stiffness coefficients used for continuum modeling do not represent stiffness of an isolated grain-pair, rather they represent a collective behavior from the grain-pair and its surroundings. An advantage of the granular micromechanics method is that the grains locations and their contacts are not needed. Conversely, for discrete modeling, one needs exact data about grains’ positions and contacts in addition to their contact properties, which can be very complicated, perhaps intractable, for real materials and complex grain assemblies.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This research is supported in part by the United States National Science Foundation grant CMMI-1068528.
References
Alibert, J. J., Seppecher, P., and Dell’Isola, F. (2003). “Truss modular beams with deformation energy depending on higher displacement gradients.” Math. Mech. Solids, 8(1), 51–73.
Altenbach, H., Eremeyev, V. A., Lebedev, L. P., and Rendon, L. A. (2010). “Acceleration waves and ellipticity in thermoelastic micropolar media.” Arch. Appl. Mech., 80(3), 217–227.
Andreaus, U., and Baragatti, P. (2011). “Cracked beam identification by numerically analysing the nonlinear behaviour of the harmonically forced response.” J. Sound Vib., 330(4), 721–742.
Andreaus, U., and Baragatti, P. (2012). “Experimental damage detection of cracked beams by using nonlinear characteristics of forced response.” Mech. Syst. Signal Process., 31, 382–404.
Andreaus, U., Chiaia, B., and Placidi, L. (2013a). “Soft-impact dynamics of deformable bodies.” Continuum Mech. Thermodyn., 25(2–4), 375–398.
Andreaus, U., Colloca, M., and Iacoviello, D. (2012). “An optimal control procedure for bone adaptation under mechanical stimulus.” Control Eng. Pract., 20(6), 575–583.
Andreaus, U., Colloca, M., and Iacoviello, D. (2014). “Optimal bone density distributions: numerical analysis of the osteocyte spatial influence in bone remodeling.” Comput. Methods Programs Biomed., 113(1), 80–91.
Andreaus, U., Giorgio, I., and Madeo, A. (2015). “Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids.” J. Appl. Math. Phys. (ZAMP), 66(1), 209–237.
Andreaus, U., Placidi, L., and Rega, G. (2013b). “Microcantilever dynamics in tapping mode atomic force microscopy via higher eigenmodes analysis.” J. Appl. Phys., 113(22), 224302.
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., and Rosi, G. (2013). “Analytical continuum mechanics a la Hamilton-Piola least action principle for second gradient continua and capillary fluids.” Math. Mech. Solids, 20(4), 375–417.
Cauchy, A.-L. (1826). “Sur l’equilibre et le mouvement d’un systeme de points materiels sollicites par des forces d’attraction ou de repulsion mutuelle.” Excercises de Mathematiques, 3, 188–212.
Chang, C. S., and Misra, A. (1989). “Computer simulation and modelling of mechanical properties of particulates.” Comput. Geotech., 7(4), 269–287.
Chang, C. S., Misra, A., and Sundaram, S. S. (1990). “Micromechanical modeling of cemented sands under low amplitude oscillations.” Geotechnique, 40(2), 251–263.
Chen, Y., and Lee, J. D. (2003). “Connecting molecular dynamics to micromorphic theory. (I). Instantaneous and averaged mechanical variables.” Physica A, 322, 359–376.
Cosserat, E., and Cosserat, F. (1909). Theory of deformable bodies, Scientific Library A. Hermann and Sons, Paris.
dell’Isola, F., Andreaus, U., and Placidi, L. (2013). “At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola.” Math. Mech. Solids, 20(8), 887–928.
dell’Isola, F., Madeo, A., and Placidi, L. (2012). “Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua.” Zamm-Zeitschrift Fur Angewandte Mathematik Und Mechanik, 92(1), 52–71.
dell’Isola, F., and Vidoli, S. (1998). “Continuum modelling of piezoelectromechanical truss beams: An application to vibration damping.” Arch. Appl. Mech., 68(1), 1–19.
Eringen, A. C. (1999). Microcontinuum field theories: Foundations and solids, Springer, New York.
Ferretti, M., Madeo, A., dell’Isola, F., and Boisse, P. (2014). “Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory.” Zeitschrift für angewandte Mathematik und Physik, 65(3), 587–612.
Ganghoffer, J.-F. (2010). “On the generalized virial theorem and Eshelby tensors.” Int. J. Solids Struct., 47(9), 1209–1220.
Germain, P. (1973). “Method of virtual power in continuum mechanics. 2. Microstructure.” Siam J. Appl. Math., 25(3), 556–575.
Greco, L., Impollonia, N., and Cuomo, M. (2014). “A procedure for the static analysis of cable structures following elastic catenary theory.” Int. J. Solids Struct., 51(7–8), 1521–1533.
Green, A. E., and Rivlin, R. S. (1964). “Multipolar continuum mechanics.” Arch. Ration. Mech. Anal., 17(2), 113–147.
Hill, R. (1952). “The elastic behaviour of a crystalline aggregate.” Proc. Phys. Soc. Sect. A, 65(5), 349.
Jenkins, J., Johnson, D., La Ragione, L., and Makse, H. (2005). “Fluctuations and the effective moduli of an isotropic, random aggregate of identical, frictionless spheres.” J. Mech. Phys. Solids, 53(1), 197–225.
Madeo, A., Neff, P., Ghiba, I.-D., Placidi, L., and Rosi, G. (2013). “Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps.” Continuum Mech. Thermodyn., 27, 551–570.
Madeo, A., Placidi, L., and Rosi, G. (2014). “Towards the design of metamaterials with enhanced damage sensitivity: Second gradient porous materials.” Res. Nondestr. Eval., 25(2), 99–124.
Maugin, G. A. (2015). “Some remarks on generalized continuum mechanics.” Math. Mech. Solids, 20(3), 280–291.
Maurini, C., dell’Isola, F., and Pouget, J. (2004). “On models of layered piezoelectric beams for passive vibration control.” J. De Physique Iv, 115, 307–316.
Maurini, C., Pouget, J., and dell’Isola, F. (2006). “Extension of the Euler-Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach.” Comput. Struct., 84(22–23), 1438–1458.
Mindlin, R. D. (1964). “Micro-structure in linear elasticity.” Arch. Ration. Mech. Anal., 16(1), 51–78.
Misra, A. (1998). “Particle kinematics in sheared rod assemblies.” Physics of Dry Granular Media, Springer, Dordrecht, Netherlands, 261–266.
Misra, A., and Chang, C. S. (1993). “Effective elastic moduli of heterogeneous granular solids.” Int. J. Solids Struct., 30(18), 2547–2566.
Misra, A., and Jiang, H. (1997). “Measured kinematic fields in the biaxial shear of granular materials.” Comput. Geotech., 20(3–4), 267–285.
Misra, A., and Poorsolhjouy, P. (2015a). “Granular micromechanics based micromorphic model predicts frequency band gaps.” Continuum Mech. Thermodyn., 1–20.
Misra, A., and Poorsolhjouy, P. (2015b). “Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics.” Math. Mech. Solids, in press.
Misra, A., and Poorsolhjouy, P. (2015c). “Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics.” Math. Mech. Complex Syst., 3(3), 285–308.
Misra, A., and Singh, V. (2014). “Nonlinear granular micromechanics model for multi-axial rate-dependent behavior.” Int. J. Solids Struct., 51(13), 2272–2282.
Misra, A., and Singh, V. (2015). “Thermomechanics based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model.” Continuum Mech. Thermodyn., 27(4–5), 787–817.
Navier, C. L. (1827). “Sur les lois de l’equilibre et du mouvement des corps solides elastiques.” Memoire de l’Academie Royale de Sciences, 7, 375–393.
Peters, J. F., and Walizer, L. E. (2013). “Patterned nonaffine motion in granular media.” J. Eng. Mech., 139(10), 1479–1490.
Placidi, L. (2014). “A variational approach for a nonlinear 1-dimensional second gradient continuum damage model.” Continuum Mech. Thermodyn., 27, 623–638.
Placidi, L., Rosi, G., Giorgio, I., and Madeo, A. (2014). “Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials.” Math. Mech. Solids, 19(5), 555–578.
Porfiri, M., dell’Isola, F., and Santini, E. (2005). “Modeling and design of passive electric networks interconnecting piezoelectric transducers for distributed vibration control.” Int. J. Appl. Electromagnet. Mech., 21(2), 69–87.
Seppecher, P., Alibert, J.-J., and dell’Isola, F. (2011). “Linear elastic trusses leading to continua with exotic mechanical interactions” J. Phys. Conf. Ser., 319(1), 012018.
Toupin, R. A. (1964). “Theories of elasticity with couple-stress.” Arch. Ration. Mech. Anal., 17(2), 85–112.
Trentadue, F. (2001). “A micromechanical model for a non-linear elastic granular material based on local equilibrium conditions.” Int. J. Solids Struct., 38(40), 7319–7342.
Vidoli, S., and dell’Isola, F. (2001). “Vibration control in plates by uniformly distributed PZT actuators interconnected via electric networks.” Eur. J. Mech. A. Solids, 20(3), 435–456.
Yang, Y., Ching, W.-Y., and Misra, A. (2011). “Higher-order continuum theory applied to fracture simulation of nano-scale intergranular glassy film.” J. Nanomech. Micromech., 1(2), 60–71.
Yang, Y., and Misra, A. (2012). “Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity.” Int. J. Solids Struct., 49(18), 2500–2514.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 143 • Issue 1 • January 2017
Copyright
© 2016 American Society of Civil Engineers.
History
Received: May 4, 2015
Accepted: Nov 20, 2015
Published online: Jan 14, 2016
Discussion open until: Jun 14, 2016
Published in print: Jan 1, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.