Abstract
The elastoviscoplasticity theory of micromorphic media at finite deformation is presented in this chapter. Micromechanical considerations are then put forward to motivate the existence of the microdeformation degrees of freedom in the case of composite materials. Mixtures of micromorphic media are finally considered with a view to homogenising the size–dependent properties of metal polycrystals.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Bibliography
E.C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326–330, 1984.
J. Altenbach, H. Altenbach, and V.A. Eremeyev. On generalized cosserattype theories of plates and shells: a short review and bibliography. Arch. Appl. Mech., 80:73–92, 2010.
A. Anthoine. Second-order homogenisation of functionally graded materials. Int. J. Solids Structures, 47:1477–1489, 2010.
O. Aslan, N. M. Cordero, A. Gaubert, and S. Forest. Micromorphic approach to single crystal plasticity and damage. International Journal of Engineering Science, 49:1311–1325, 2011.
N. Auffray, R. Bouchet, and Y. Bréchet. Derivation of anisotropic matrix for bi–dimensional strain–gradient elasticity behavior. International Journal of Solids and Structures, 46:440–454, 2009.
N. Auffray, R. Bouchet, and Y. Bréchet. Strain gradient elastic homogenization of bidimensional cellular media. International Journal of Solids and Structures, 47:1698–1710, 2010.
V.P. Bennett and D.L. McDowell. Crack tip displacements of microstructurally small surface cracks in single phase ductile polycrystals. Engineering Fracture Mechanics, 70(2):185–207, 2003.
D. Besdo. Towards a Cosserat-theory describing motion of an originally rectangular structure of blocks. Arch. Appl. Mech., 80:25–45, 2010.
J. Besson, G. Cailletaud, J.-L. Chaboche, S. Forest, and M. Blétry. Non–Linear Mechanics of Materials. Series: Solid Mechanics and Its Applications, Vol. 167, Springer, ISBN: 978-90-481-3355-0, 433 p., 2009.
C. Boutin. Microstructural effects in elastic composites. Int. J. Solids Structures, 33:1023–1051, 1996.
F. Bouyge, I. Jasiuk, and M. Ostoja-Starzewski. A micromechanically based couple-stress model of an elastic two-phase composite. Int. J. Solids Structures, 38:1721–1735, 2001.
F. Bouyge, I. Jasiuk, S. Boccara, and M. Ostoja-Starzewski. A micromechanically based couple-stress model of an elastic orthotropic two-phase composite. European Journal of Mechanics A/solids, 21:465–481, 2002.
H. Chen, X. Liu, G. Hu, and H. Yuan. Identification of material parameters of micropolar theory for composites by homogenization method. Computational Materials Science, 46:733–737, 2009.
N. M. Cordero, S. Forest, E. P. Busso, S. Berbenni, and M. Cherkaoui. Grain size effect in generalized continuum crystal plasticity. In I. R. Ionescu, S. Bouvier, O. Cazacu, and P. Franciosi, editors, Plasticity of crystalline materials, pages 101–122. ISTE-Wiley, 2011.
N. M. Cordero, S. Forest, E. P. Busso, S. Berbenni, and M. Cherkaoui. Grain size effects on plastic strain and dislocation density tensor fields in metal polycrystals. Computational Materials Science, 52:7–13, 2012.
N.M. Cordero, A. Gaubert, S. Forest, E. Busso, F. Gallerneau, and S. Kruch. Size effects in generalised continuum crystal plasticity for two–phase laminates. Journal of the Mechanics and Physics of Solids, 58:1963–1994, 2010.
T. Dillard, S. Forest, and P. Ienny. Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. European Journal of Mechanics A/Solids, 25:526–549, 2006.
K. Enakoutsa and J.B. Leblond. Numerical implementation and assessment of the glpd micromorphic model of ductile rupture. European Journal of Mechanics A/solids, 28:445–460, 2009.
A.C. Eringen. Theory of thermo-microstretch elastic solids. Int. J. Engng Sci., 28:1291–1301, 1990.
A.C. Eringen. Microcontinuum field theories. Springer, New York, 1999.
A.C. Eringen and E.S. Suhubi. Nonlinear theory of simple microelastic solids. Int. J. Engng Sci., 2:189–203, 389–404, 1964.
F. Feyel. A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua. Comp. Meth. Appl. Mech. Engng, 192:3233–3244, 2003.
S. Forest. The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics, 135:117–131, 2009.
S. Forest. Mechanics of generalized continua : Construction by homogenization. Journal de Physique IV, 8:Pr4–39–48, 1998.
S. Forest. Homogenization methods and the mechanics of generalized continua–Part 2. Theoretical and Applied Mechanics, 28–29:113–143, 2002.
S. Forest. Aufbau und Identifikation von Stoffgleichungen für höhere Kontinua mittels Homogenisierungsmethoden. Technische Mechanik, Band 19, Heft 4:297–306, 1999.
S. Forest. Cosserat media. In K.H.J. Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, and S. Mahajan, editors, Encyclopedia of Materials : Science and Technology, pages 1715–1718. Elsevier, 2001.
S. Forest and A. Bertram. Formulations of strain gradient plasticity. In H. Altenbach, G. A. Maugin, and V. Erofeev, editors, Mechanics of Generalized Continua, pages 137–150. Advanced Structured Materials vol. 7, Springer, 2011.
S. Forest and K. Sab. Cosserat overall modeling of heterogeneous materials. Mechanics Research Communications, 25(4):449–454, 1998.
S. Forest and R. Sievert. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mechanica, 160:71–111, 2003.
S. Forest and R. Sievert. Nonlinear microstrain theories. International Journal of Solids and Structures, 43:7224–7245, 2006.
S. Forest and D. K. Trinh. Generalized continua and non–homogeneous boundary conditions in homogenization methods. ZAMM, 91:90–109, 2011.
S. Forest, F. Pradel, and K. Sab. Asymptotic analysis of heterogeneous Cosserat media. International Journal of Solids and Structures, 38:4585–4608, 2001.
M.G.D. Geers, V.G. Kouznetsova, and W.A.M Brekelmans. Gradient–enhanced computational homogenization for the micro–macro scale transition. Journal de Physique IV, 11:Pr5–145–152, 2001.
P. Germain. The method of virtual power in continuum mechanics. part 2 : Microstructure. SIAM J. Appl. Math., 25:556–575, 1973.
J.D. Goddard. Mathematical models of granular matter, ed. by P. Mariano, G. Capriz, and P. Giovine, chapter From Granular Matter to Generalized Continuum, pages 1–20. vol. 1937 of Lecture Notes in Mathematics, Springer, Berlin, 2008.
M. Gologanu, J. B. Leblond, and J. Devaux. Continuum micromechanics, volume 377, chapter Recent extensions of Gurson’s model for porous ductile metals, pages 61–130. Springer Verlag, CISM Courses and Lectures No. 377, 1997.
M.A. Goodman and S.C. Cowin. A continuum theory for granular materials. Arch. Rational Mech. and Anal., 44:249–266, 1972.
P. Grammenoudis and Ch. Tsakmakis. Micromorphic continuum Part I: Strain and stress tensors and their associated rates. International Journal of Non–Linear Mechanics, 44:943–956, 2009.
P. Grammenoudis, Ch. Tsakmakis, and D. Hofer. Micromorphic continuum Part II: Finite deformation plasticity coupled with damage. International Journal of Non–Linear Mechanics, 44:957–974, 2009.
C.B. Hirschberger, E. Kuhl, and P. Steinmann. On deformational and configurational mechanics of micromorphic hyperelasticity - theory and computation. Computer Methods in Applied Mechanics and Engineering, 196:4027–4044, 2007.
R. Jänicke and Diebels. A numerical homogenisation strategy for micromorphic continua. Nuovo Cimento della Societa Italiana di Fisica C-Geophysics and Space Physics, 32:121–132, 2009.
R. Jänicke, S. Diebels, H.-G. Sehlhorst, and A. Düster. Two–scale modelling of micromorphic continua. Continuum Mechanics and Thermodynamics, 21:297–315, 2009.
C.B. Kafadar and A.C. Eringen. Micropolar media: I the classical theory. Int. J. Engng Sci., 9:271–305, 1971.
T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin. Determination of the size of the representative volume element for random composites : statistical and numerical approach. International Journal of Solids and Structures, 40:3647–3679, 2003.
N. Kirchner and P. Steinmann. A unifying treatise on variational principles for gradient and micromorphic continua. Philosophical Magazine, 85: 3875–3895, 2005.
V. G. Kouznetsova, M. G. D. Geers, and W. A. M. Brekelmans. Multiscale constitutive modelling of heterogeneous materials with a gradientenhanced computational homogenization scheme. Int. J. Numer. Meth. Engng, 54:1235–1260, 2002.
V. G. Kouznetsova, M. G. D. Geers, and W. A. M. Brekelmans. Multi-scale second-order computational homogenization of multi-phase materials : A nested finite element solution strategy. Computer Methods in Applied Mechanics and Engineering, 193:5525–5550, 2004.
S. Kruch and S. Forest. Computation of coarse grain structures using a homogeneous equivalent medium. Journal de Physique IV, 8:Pr8–197–205, 1998.
M. Lazar and G.A. Maugin. On microcontinuum field theories: the eshelby stress tensor and incompatibility conditions. Philosophical Magazine, 87: 3853–3870, 2007.
X. Liu and G. Hu. Inclusion problem of microstretch continuum. International Journal of Engineering Science, 42:849–860, 2003.
J. Mandel. Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Structures, 9:725–740, 1973.
G.A. Maugin. Internal variables and dissipative structures. J. Non–Equilib. Thermodyn., 15:173–192, 1990.
R.D. Mindlin. Micro–structure in linear elasticity. Arch. Rat. Mech. Anal., 16:51–78, 1964.
R.D. Mindlin. Second gradient of strain and surface–tension in linear elasticity. Int. J. Solids Structures, 1:417–438, 1965.
M. Ostoja-Starzewski, S. D. Boccara, and I. Jasiuk. Couple-stress moduli and characteristic length of two-phase composite. Mechanics Research Communication, 26:387–396, 1999.
R.A. Regueiro. On finite strain micromorphic elastoplasticity. International Journal of Solids and Structures, 47:786–800, 2010.
G. Salerno and F. de Felice. Continuum modeling of periodic brickwork. International Journal of Solids and Structures, 46:1251–1267, 2009.
E. Sanchez-Palencia. Comportement local et macroscopique d’un type de milieux physiques hétérogènes. International Journal of Engineering Science, 12:331–351, 1974.
V. Sansalone, P. Trovalusci, and F. Cleri. Multiscale modeling of materials by a multifield approach : microscopic stress and strain distribution in fiber-matrix composites. Acta Materialia, 54:3485–3492, 2006.
C. Sansour. A unified concept of elastic–viscoplastic Cosserat and micromorphic continua. Journal de Physique IV, 8:Pr8–341–348, 1998a.
C. Sansour. A theory of the elastic–viscoplastic cosserat continuum. Archives of Mechanics, 50:577–597, 1998b.
C. Sansour, S. Skatulla, and H. Zbib. A formulation for the micromorphic continuum at finite inelastic strains. Int. J. Solids Structures, 47:1546–1554, 2010.
H. Steeb and S. Diebels. A thermodynamic–consistent model describing growth and remodeling phenomena. Computational Materials Science, 28:597–607, 2003.
P. Trovalusci and R. Masiani. Non-linear micropolar and classical continua for anisotropic discontinuous materials. International Journal of Solids and Structures, 40:1281–1297, 2003.
C. Truesdell and W. Noll. The non-linear field theories of mechanics. Handbuch der Physik, edited by S. Flügge, reedition Springer Verlag 2004, 1965.
C.A. Truesdell and R.A. Toupin. Handbuch der Physik, S. Flügge, editor, vol. 3, chapter The classical field theories, pages 226–793. Springer verlag, Berlin, 1960.
F. Xun, G. Hu, and Z. Huang. Size–dependence of overall in–plane plasticity for fiber composites. International Journal of Solids and Structures, 41: 4713–4730, 2004.
X. Yuan, Y. Tomita, and T. Andou. A micromechanical approach of nonlocal modeling for media with periodic microstructures. Mechanics Research Communications, 35:126–133, 2008.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 CISM, Udine
About this chapter
Cite this chapter
Forest, S. (2013). Micromorphic Media. In: Altenbach, H., Eremeyev, V.A. (eds) Generalized Continua from the Theory to Engineering Applications. CISM International Centre for Mechanical Sciences, vol 541. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1371-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-7091-1371-4_5
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-1370-7
Online ISBN: 978-3-7091-1371-4
eBook Packages: EngineeringEngineering (R0)