Abstract
The traditional principles of the theory of constitutive relations in classical continuum mechanics are discussed. In light of the approaches by A. A. Ilyushin and by W. Noll, the equivalence and the completeness of their general reduced forms of constitutive relations for simple classical media are noted. First, it is mentioned that a possible presence of internal kinematic constraints is rarely taken into account and needs special modifications in formulating the principles and relations. Secondly, a systematic study of internal body forces in constitutive relations has not been done before. Here a unified approach to the theory of constitutive relations is proposed to describe properties of deformation resistance of a body including both internal kinematic constraints and internal body forces. The general reduced forms of the system of constitutive relations are derived in terms of different definitions of a dynamical process in a body. The case of a simple body is considered in detail in view of the theory of objective tensors, their diagrams and frame-independent relations between objective tensors. The completeness of Ilyushin’s and Noll’s types of relations for the most general constitutive formulations is noted and confirmed by examples.
Similar content being viewed by others
References
Ilyushin AA (1948) Plastichnost’. Uprugo-plasticheskiye deformatsii (Plasticity. Elastic-plastic deformations). Gostekhizdat, Moscow-Leningrad (see also Iliouchine A (1956) Plasticite. Eyrolles, Paris)
Ilyushin AA (1954) On relations between stresses and small strains in continuum mechanics. Appl Math Mech 18(6): 641–666
Ilyushin AA (1963) Plastichnost’. Osnovy obshchey matematicheskoy teorii (Plasticity. Foundations of the General Mathematical Theory). Academy of Sciences of the USSR, Moscow
Ilyushin AA (1971) Mekhanika sploshnoy sredy (Continuum mechanics). Moscow State University Press, Moscow, 1971 (see also Second Edition, 1978. Third Edition, 1990)
Ilyushin AA (1994) Functionals and irrevesibility measures on sets of processes in continuum mechanics (CM). Trans Acad Sci USSR 337(1): 48–50
Ilyushin AA (1996) Non-symmetry of strain and stress tensors in continuum mechanics. Vestnik Moskovskogo Universiteta (Bulletin of Moscow University), Ser 1. Mathematics. Mechanics 5: 6–14
Ilyushin AA, Ilyushina GA (1983) Problems of thermodynamics of irreversible processes. Vestnik Moskovskogo Universiteta (Bulletin of Moscow University), Ser 1. Mathematics. Mechanics 3: 73–80
Ilyushin AA, Lensky VS (1959) Soprotivleniye materialov (Strength of materials). Fizmatgiz, Moscow (see also Ilyushin AA, Lensky VS (1967) Strength of materials. Pergamon, New York
Ilyushin AA, Lomakin VA (1971) Moment theories in mechanics of solids. Strength and plasticity. Nauka, Moscow, pp 54–60
Ilyushin AA, Pobedria BE (1970) Osnovy matematicheskoy teorii termovyazkouprugosti (Foundations of mathematical theory of thermoviscoelasticity). Nauka, Moscow
Ilyushin AA (2003) Izbrannye trudy. Tom 1 (1935–1945) (Selected papers, vol 1 (1935–1945)). Fizmatlit, Moscow
Ilyushin AA (2004) Izbrannye trudy. Tom 2 (1946–1966) Plastichnost’ (Selected papers (1946–1966), vol 2. Plasticity). Fizmatlit, Moscow
Ilyushin AA (2007) Izbrannye trudy. Tom 3 Teoriya termovyazkouprugosti (Selected papers, vol 3. Theory of thermoviscoelasticity). Fizmatlit, Moscow
Ilyushin AA (2009) Izbrannye trudy. Tom 4 Modelirovanie dinamicheskikh processov v tverdykh telakh i inzhenernye prilozheniya (Selected papers Vol 4. Modelling of dynamic processes in solids and engineeric applications). Fizmatlit, Moscow
Noll W (1958) A mathematical theory of the mechanical behavior of continuous media. Arch Rat Mech Anal 2: 197–226
Truesdell C (1972) A first course in rational continuum mechanics. The Johns Hopkins University Press, Baltimore
Truesdell C, Noll W (1965) The nonlinear field theories of mechanics. Encyclopedia of physics, vol III/3. Springer-Verlag, Berlin (see also Second Edition, 1992. Third Edition, 2004)
Noll W (1967) Materially uniform simple bodies with inhomogeneities. Arch Rat Mech Anal 27: 1–32
Noll W (1972) A new mathematical theory of simple materials. Arch Rat Mech Anal 48: 1–50
Noll W (1973) Lectures on the foundations of continuum mechanics and thermodynamics. Arch Rat Mech Anal 52: 62–92
Antman SS (2005) Nonlinear problems of elasticity. Springer, New York
Astarita G, Marrucci G (1974) Principles of non-newtonian fluid mechanics. McGraw-Hill, New York
Ciarlet PG (1988) Mathematical elasticity. North-Holland, Amsterdam
Germain P (1973) Course de Mécanique des Millieux Continus. Théorie Générale. Masson, Paris
Gurtin ME (1981) An introduction to continuum mechanics. Academic Press, New York
Levitas VI (1987) Bol’shie uprugo-plasticheskie deformacii materialov pri vysokom davlenii (Large elastoplastic deformations of materials under high pressure). Naukova Dumka, Kiev
Lourie AI (1980) Nelineynaya teoriya uprugosti (Nonlinear theory of elasticity). Nauka, Moscow
Maugin GA (1992) The thermomechanics of plasticity and fracture. Cambridge University Press, Cambridge
Mueller I (1984) Thermodynamics. Pitman, Boston
Pobedria BE, Georgievskii DV (2006) Osnovy mekhaniki sploshnoi sredy (Foundations of continuum mechanics). Fizmatlit, Moscow
Pozdeyev AA, Trusov PV, Nyashin YuI (1986) Bol’shie upugo-plasticheskie deformacii (Large elastoplastic deformations). Nauka, Moscow
Silhavy M (1997) The mechanics and thermodynamics of continuous media. Springer, Berlin
Wilmanski K (1998) Thermomechanics of continua. Springer, Berlin
Antman SS, Marlow RS (1991) Material constraints, Lagrange multipliers, and compatibility. Arch Rat Mech Anal 116: 257–299
Gurtin ME, Podio-Guidugli P (1973) The thermodynamics of constrained materials. Arch Rat Mech Anal 51: 192–208
Podio-Guidugli P (2000) The compatibility constraint in linear elasticity. J Elast 59: 393–398
Podio-Guidugli P, Vianello M (1989) Constraint manifolds for isotropic solids. Arch Rat Mech Anal 105: 105–121
Sedov LI (1973) Mekhanika sploshnoy sredy (Continuum mechanics). Nauka, Moscow
DeSimone A, Podio-Guidugli P (1995) Inertial and self interactions in structured continua: liquid crystals and magnetostrictive solids. Meccanica 30: 629–640
Sobolev SL (1989) Izbrannye voprosy teorii funkcional’nykh prostranstv i obobshchennykh funkciy (Selected problems of the theory of functional spaces and distributions). Nauka, Moscow
Ogden RW (1984) On Eulerian and Lagrangean objectivity in continuum mechanics. Arch Mech 36: 207–218
Brovko GL (1996) Development of the mathematical apparatus and the foundations of the general theory of constitutive relations in continuum mechanics. DSci Dissertation, Moscow University
Brovko GL (2006) Problems of invariance in classical and nonclassical models of continua. In: Kiyko IA, Vasin RA, Brovko GL (eds) Elasticity and anelasticity. URSS, Moscow, pp 110–123
Brovko GL (2009) Effective invariance properties of processes and relations in continuum mechanics. In: Modern problems of mathematics and mechanics. Vol 2. Mechanics. Issue 1. Moscow University Press, Moscow, pp 108–126
Brovko GL (1990) Material and spatial representations of the constitutive relations of deformable media. J Appl Math Mech 54: 668–677
Cotter BA, Rivlin RS (1955) Tensors assotiated with time-dependent stress. Quart Appl Math 3: 177–188
Dienes JK (1979) On the analysis of rotation and stress rate in deforming bodies. Acta Mech 32: 217–232
Jaumann G (1911) Geschlossenes system physikalischer und chemischer differenzialgesetze. Sitzber Akad Wiss Wien 120: 385–530
Oldroyd JG (1950) On the formulation of rheological equations of state. Proc R Soc Lond A 200: 523–541
Sedov LI (1960) Different definitions of the rate of change of a tensor. J Appl Math Mech 24: 579–586
Truesdell C (1955) Hypo-elasticity. J Rat Mech Anal 4: 83–133
Zaremba S (1903) Sur une forme perfectionee de la theorie de la relaxation. Bull Acad Sci Cracovie 594–614
Brovko GL (1989) Notions of process image and five-dimensional isotropy of material properties at finite strains. Doklady Akademii Nauk SSSR 308(3): 565–570
Bell JF (1985) Contemporary perspectives in finite strain plasticity. Int J Plast 1: 3–27
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brovko, G.L. On general principles of the theory of constitutive relations in classical continuum mechanics. J Eng Math 78, 37–53 (2013). https://doi.org/10.1007/s10665-011-9508-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-011-9508-y