Abstract
Structured deformations are used to refine the basic ingredients of continuum field theories and to derive a system of field equations for elastic bodies undergoing submacroscopically smooth geometrical changes as well as submacroscopically non-smooth geometrical changes (disarrangements). The constitutive assumptions employed in this derivation permit the body to store energy as well as to dissipate energy in smooth dynamical processes. Only one non-classical field G, the deformation without disarrangements, appears in the field equations, and a consistency relation based on a decomposition of the Piola-Kirchhoff stress circumvents the use of additional balance laws or phenomenological evolution laws to restrict G. The field equations are applied to an elastic body whose free energy depends only upon the volume fraction for the structured deformation. Existence is established of two universal phases, a spherical phase and an elongated phase, whose volume fractions are (1−γ0)3 and (1−γ0) respectively, with γ0:=(\( \sqrt 5 \)−1)/2 the “golden mean”.
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References
G. Del Piero and D.R. Owen, Structured deformations of continua. Arch. Rational Mech. Anal. 124 (1993) 99–155.
G. Del Piero and D.R. Owen, Integral-gradient formulae for structured deformations. Arch. Rational Mech. Anal. 131 (1995) 121–138.
G. Del Piero and D.R. Owen, Structured Deformations. Quaderni dell' Istituto Nazionale di Alta Matematica, Gruppo Nazionale di Fisica Matematica No. 58 (2000).
D.R. Owen, Twin balance laws for bodies undergoing structured motions. In: P. Podio-Guidugli and M. Brocato (eds), Rational Continua, Classical and New. Springer-Verlag, New York (2002); Research Report No. 01-CNA-005, February 2001, Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University.
D.R. Owen and R. Paroni, Second order structured deformations. Arch. Rational Mech. Anal. 155 (2000) 215–235.
R. Choksi and I. Fonseca, Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal. 138 (1997) 37–103.
R. Choksi, G. Del Piero, I. Fonseca and D.R. Owen, Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999) 321–356.
L. Deseri and D.R. Owen, Energetics of two-level shears and hardening of single crystals. Mathematics and Mechanics of Solids 7 (2002) 113–147.
G. Del Piero, The energy of a one-dimensional structured deformation. Mathematics and Mechanics of Solids 6 (2001) 387–408.
G. Capriz, Continua with Microstructure. Springer Tracts in Natural Philosophy 35. Springer-Verlag, New York (1989).
A. Eringen, Microcontinuum Field Theories, I. Foundations and Solids. Springer-Verlag, New York (1999).
M. Renardy, W. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity. Pitman Monographs and Surveys in Pure and Applied Mathematics 35. Longman Scientific and Technical (1987).
L. Deseri and D.R. Owen, Invertible structured deformations and the geometry of multiple slip in single crystals. Internat. J. Plasticity 18 (2002) 833–849.
M. Boyce, G. Weber and D. Parks, On the kinematics of finite strain plasticity. J. Mechanics and Physics of Solids 37 (1989) 647–665.
D.R. Owen, Structured deformations and the refinements of balance laws induced by microslip. Internat. J. Plasticity 14 (1998) 289–299.
P. Haupt, Continuum Mechanics and Theory of Materials. Springer-Verlag, Berlin (2000).
W. Noll, La mécanique classique, basée sur un axiome d'objectivité. In: La Méthode Axiomatique dans les Mécaniques Classiques and Nouvelles (Colloque International, Paris, 1959). Gauthier-Villars, Paris (1963) pp. 47–56.
A.E. Green and R.S. Rivlin, On Cauchy's equations of motion. J. Appl. Math. Phys. 15 (1964) 290–292.
M.E. Gurtin, An Introduction to Continuum Mechanics. Academic Press, New York (1981).
B.D. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13 (1963) 167–178.
C.M. Dafermos, Quasilinear hyperbolic systems with involutions. Arch. Rational Mech. Anal. 94 (1986) 373–389.
W. Noll, On the continuity of the solid and fluid states. J. Rational Mech. Anal. 4 (1955) 3–81.
D. Luenberger, Linear and Nonlinear Programming, 2nd edn. Addison-Wesley, Reading, MA (1989).
J.L. Ericksen, Loading devices and stability of equilibrium. In: Nonlinear Elasticity. Academic Press (1973) pp. 161–173.
M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237–277.
I. Fonseca and G. Parry, Equilibrium configurations of defective crystals. Arch. Rational Mech. Anal. 120 (1992) 245–283.
C. Davini and G. Parry, On defect-preserving deformations in crystals. Internat. J. Plasticity 5 (1989) 337–369.
V. Mizel, On the ubiquity of fracture in non-linear elasticity. J. Elasticity 52 (1999) 257–266.
M. Šilhavý and J. Kratochvíl, A theory of inelastic behavior of materials, Part I. Arch. Rational Mech. Anal. 65 (1977) 97–129; Part II. Arch. Rational Mech. Anal. 65 (1977) 131–152.
W. Noll, A new mathematical theory of simple materials. Arch. Rational Mech. Anal. 48 (1972) 1–50.
A. Bertram, An alternative approach to finite plasticity based on material isomorphisms. Internat. J. Plasticity 14 (1999) 353–374.
C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, 2nd edn. Springer-Verlag, Berlin (1992).
W. Noll, Materially uniform simple bodies with inhomogeneities. Arch. Rational Mech. Anal. 27 (1967) 1–32.
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Deseri, L., Owen, D.R. Toward a Field Theory for Elastic Bodies Undergoing Disarrangements. Journal of Elasticity 70, 197–236 (2003). https://doi.org/10.1023/B:ELAS.0000005584.22658.b3
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DOI: https://doi.org/10.1023/B:ELAS.0000005584.22658.b3