Mechanics of extended continua: modeling and simulation of elastic microstretch materials
Tóm tắt
The investigation of microstretch and micromorphic continua (which are prominent examples of so-called extended continua) dates back to Eringens pioneering works in the mid 1960, cf. (Eringen in Mechanics of micromorphic materials. Springer, Berlin Heidelberg New York, pp 131–138, 1966; Eringen in Int J Eng Sci 8:819–828; Eringen in Microcontinuum field theories. Springer, Berlin Heidelberg New York, 1999). Here, we re-derive the governing equations of microstretch continua in a variational setting, providing a natural framework within which numerical implementations of the model equations by means of the finite element method can be obtained straightforwardly. In the application of Dirichlets principle, the postulation of an appropriate form of the Helmholtz free energy turns out to be crucial to the derivation of the balance laws and constitutive relations for microstretch continua. At present, the material parameters involved in the free energy have been assigned fixed values throughout all numerical simulations—this simplification is addressed in detail as the influence of those parameters must not be underestimated. Since only few numerical results demonstrating elastic microstretch material behavior in engineering applications are available, the focus is here on the presentation of numerical results for simple twodimensional test specimens subjected to a plane strain condition and uniaxial tension. Confidence in the simulations for microstretch materials is gained by showing that they exhibit a “downward-compatibility” to Cosserat continuum formulation: by switching off all stretch-related effects, the governing set of equations reduces to the one used for polar materials. Further, certain material parameters can be chosen to act as penalty parameters, forcing stretch-related contributions to an almost negligible range in a full microstretch model so that numerical results obtained for a polar model can be obtained as a limiting case from the full microstretch model.
Tài liệu tham khảo
Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mat Tech 106:326–330
Allen SJ, DeSilva CN, Kline KA (1967) Theory of simple deformable directed fluids. Phys Fluids 10(12):2551–2555
Braess D (2001) Finite elements. theory, fast solvers and applications in solid mechanics. Cambridge University Press, London
Brenner SC, Scott LR (2002) The mathematical theory of finite element methods, 2nd edn. Springer, Berlin Heidelberg New York
Ciarletta M (1999) On the bending of microstretch elastic plates. Int J Eng Sci 37(10):1309–1318
Cosserat E, Cosserat F (1909) Théorie des corps déformable. Hermann et Fils
Cowin SC (1974) The theory of polar fluids. Adv Appl Mech 14:279–347
deBorst R (1991) Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng Comput 8:317–332
DeCicco S (2003) Stress concentration effects in microstretch elastic bodies. Int J Eng Sci 41(2):187–199
Diebels S (1999) A micropolar theory of porous media: constitutive modelling. Transport Porous Media 34:193–208
Dietsche A, Steinmann P, Willam K (1993) Micropolar elastoplasticity and its role in localization analysis. Int J Plast 9:813–831
Eringen AC (1966) Mechanics of micromorphic materials. In: Görtler H, Sorger P (eds) Proceedings of 11th international congress of applied mechanics. Springer, Berlin Heidelberg New York, pp 131–138
Eringen AC (1968) Theory of micropolar elasticity. In: Liebowitz H (eds) Fracture An advanced treatise, vol II. Academic, New York, pp 621–729
Eringen AC (1970) Balance laws of micromorphic mechanics. Int J Eng Sci 8:819–828
Eringen AC (1972) Theory of micromorphic materials with memory. Int J Eng Sci 10:623–641
Eringen AC (1990) Theory of thermo-microstretch elastic solids. Int J Eng Sci 28(12):1291–1301
Eringen AC (1992) Balance laws of micromorphic continua revisited. Int J Eng Sci 30(6):805–810
Eringen AC (1999) Microcontinuum field theories. Springer, Berlin Heidelberg New York
Eringen AC, Kafadar CB (1976) Polar field theories. In: Continuum physics, vol 4. Eringen AC (eds) Academic, New York, pp 1–73
Forest S, Sievert R (2003) Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech 160:71–111
Günther W (1958) Zur Statik und Kinematik des Cosserat’schen Kontinuums. Abh. Braunschweigische Wiss. Gesell. 10:195–213
Green AE, Rivlin RS (1964) Multipolar continuum mechanics. Arch Ration Mech Anal 17:113–147
Hill R (1998) The mathematical theory of plasticity. Clarendon, Oxford
Iesan D, Nappa L (2001) On the plane strain of microstretch elastic solids. Int J Eng Sci 39:1815–1835
Iesan D, Nappa L (2001) Extremum principles and existence results in micromorphic elasticity. Int J Eng Sci 39:2051–2070
Iesan D, Quintanilla R (1994) Existence and continuous dependence results in the theory of microstretch elastic bodies. Int J Eng Sci 32:991–1001
Iesan D, Scalia A (2003) On complex potentials in the theory of microstretch elastic bodies. Int J Eng Sci 41(17):1989–2003
KirchnerN, Steinmann P (2005) A unifying treatise on variational principles for gradient and micro-morphic continua. Philosophical Magazine 85(33–35):3875–3895
Kumar R, Deswal S (2002) Wave propagation through a cylindrical bore contained in a microstretch elastic medium. J Sound Vib 250(4):711–722
Lax PD, Milgram AN (1954) Parabolic equations. Contributions to the theory of partial differential equations. Ann Math Stud 33:167–190
Lee JD, Chen Y (2003) Constitutive relations of micromorphic thermoplasticity. Int J Eng Sci 41(17):387–399
Leslie FM (1968) Some constitutive equations for liquid crystals. Arch Ration Mech Anal 28:265–283
Marsden JE, Hughes JR, Hughes TJR (1983) Mathematical Foundations of Elasticity. Prentice-Hall, Englewood Cliffs
Mindlin RD (1963) Microstructure in linear elasticity. Arch Ration Mech Anal 16:51–78
Mühlhaus HB (1989) Application of Cosserat theory in numerical solutions of limit load problems. Ing Arch 59:124–137
Mühlhaus HB, Vardoulakis I (1987) The thickness of shear bands in granular materials. Géotechnique 37:271–283
Nappa L (2001) Variational principles in micromorphic elasticity. Mech Res Commun 28:405–412
Neff P (2006) Existence of minimizers for a finite-strain micromorphic elastic solid. Proc R Soc Edinburgh, A 136:997–1012
Oden JT, Reddy JN (1976) Variational methods in theoretical mechanics. Springer, Berlin Heidelberg New York
Scalia A (2000) Extension, bending and torsion of anisotropic microstretch elastic cylinders. Math Mech Solids 5(1):31–40
Schwab C (1998) p- and hp- finite element methods. Theory and applications in solid and fluid mechanics. Clarendon Press, Oxford
Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, Berlin Heidelberg New York
Singh B (2001) Reflection and refraction of plane waves at a liquid/thermo-microstretch elastic solid interface. Int J Eng Sci 39(5):583–598
Steeb H, Diebels S (2005) Continua with affine microstructure: theoretical aspects and applications. Proc Appl Math Mech 5:319–320
Steinmann P (1995) Theory and numerics of ductile micropolar elastoplastic damage. Int J Eng Sci 38:583–606
Steinmann P (1999) Formulation and computation of geometrically nonlinear gradient damage. Int J Numer Methods Eng 46:757–779
Toupin RA (1962) Elastic materials with couple stress. Arch Ration Mech Anal 11:385–413
Toupin RA (1964) Theory of elasticity with couple stresses. Arch Ration Mech Anal 17:85–112
Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (eds) Encyclopedia of Physics III/3. Springer, Berlin Heidelberg New York
Zienkiewicz OC, Taylor RL (1989,1991) The finite element method, 4th edn, vols 1 and 2. McGraw Hill, New York