×

zbMATH — the first resource for mathematics

Unified thermodynamic framework for nonlocal/gradient continuum theories. (English) Zbl 1032.74505
Summary: A thermodynamic framework, equipped with the concept of nonlocality (energy) residual, is utilized to address nonlocal/gradient internal variable material models. A unified procedure is provided for either nonlocal and gradient materials, which makes it possible to determine the thermodynamic restrictions upon the constitutive equations, and in particular the pertinent state equations, the consistent form of the dissipation power and the constitutive expression of the nonlocality residual. Additionally, for gradient models, the associated nonstandard boundary conditions are derived, pointing out their basically constitutive nature and their substantial differences from the standard ones. Gradient elasticity and gradient plasticity are addressed in some details. Gradient elasticity is also compared with the Toupin–Mindlin strain gradient elasticity and their remarkable differences are enumerated. A few simple examples are reported as an illustration.

MSC:
74A15 Thermodynamics in solid mechanics
74A30 Nonsimple materials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aifantis, E.C., Update on a class of gradient theories, Mech. mater., 35, 259-280, (2003)
[2] Altan, B.S.; Aifantis, E.C., On some aspects in the special theory of gradient elasticity, J. mech. behavior mater., 8, 231-282, (1997)
[3] Bažant, Z.P.; Cedolin, L., Stability of structures, (1991), Oxford University Press New York
[4] Bažant, Z.P.; Planas, J., Fracture and size effects in concrete and other quasibrittle materials, (1998), CRC Press LLC Boston
[5] Bažant, Z.P.; Jirásek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, J. engrg. mech. ASCE, 128, 1119-1149, (2002)
[6] Benvenuti, E.; Borino, G.; Tralli, A., A thermodynamically consistent nonlocal formulation of damaging materials, Eur. J. mech. A solids, 21, 535-553, (2002) · Zbl 1038.74006
[7] Borino, G., Failla, B., Parrinello, F., 2003. A symmetric nonlocal damage theory. Int. J. Solids Structures, in press · Zbl 1038.74509
[8] Borino, G.; Fuschi, P.; Polizzotto, C., A thermodynamic approach to nonlocal plasticity and related variational principles, ASME J. appl. mech., 66, 952-963, (1999)
[9] ()
[10] de Borst, R.; Mühlhaus, H.-B., Gradient dependent plasticity: formulation and algorithmic aspects, Int. J. numer. methods engrg., 35, 539-541, (1992) · Zbl 0768.73019
[11] Capurso, M., Minimum principle in elastic – plastic incremental analysis problems, Atti accad. naz. lincei (8), XLIV, 4-5, 552-560, (1969), (in Italian)
[12] Capurso, M.; Maier, G., Incremental elastoplastic analysis and quadratic optimization, Meccanica, V, 107-116, (1970) · Zbl 0198.58301
[13] Colemann, B.D.; Gurtin, M.E., Thermodynamics with internal variables, J. chem. phys., 47, 597-613, (1967)
[14] Dunn, J.E.; Serrin, J., On the thermodynamics of interstitial working, Arch. rational mech. anal., 88, 95-133, (1985) · Zbl 0582.73004
[15] Edelen, D.G.B.; Laws, N., Thermodynamics with internal variables, J. chem. phys., 47, 597-613, (1971)
[16] Edelen, D.G.B.; Green, A.E.; Laws, N., Nonlocal continuum mechanics, Arch. rational mech. anal., 43, 24-35, (1971)
[17] Eringen, A.C., Nonlocal polar elastic continua, Int. J. engrg. sci., 10, 1-16, (1972) · Zbl 0229.73006
[18] Eringen, A.C., Nonlocal micropolar field theory, (), 205-267
[19] Eringen, A.C., Line crack subjected to shear, Int. J. fracture, 14, 367-379, (1978)
[20] Eringen, A.C., Line crack subjected to antiplane shear, Engrg. fract. mech., 12, 211-219, (1979)
[21] Eringen, A.C., On nonlocal plasticity, Int. J. engrg. sci., 12, 1461-1474, (1981) · Zbl 0474.73028
[22] Eringen, A.C., Theory of nonlocal elasticity and some applications, Res. mech., 21, 313-342, (1987)
[23] Eringen, A.C.; Edelen, D.G.B., Nonlocal elasticity, Int. J. engrg. sci., 10, 233-248, (1972) · Zbl 0247.73005
[24] Germain, P.; Nguyen, Q.S.; Suquet, P., Continuum thermodynamics, ASME J. appl. mech., 50, 1010-1021, (1983) · Zbl 0536.73004
[25] Gurtin, M.E., Thermodynamics and the possibility of spatial interaction on elastic materials, Arch. rational mech. anal., 19, 339-352, (1965) · Zbl 0146.21106
[26] Kröner, E., Elasticity theory of materials with long range cohesive forces, Int. J. solids structures, 3, 731-742, (1967) · Zbl 0163.19402
[27] Lemaitre, J.; Chaboche, J.-L., Mechanics of solid materials, (1990), Cambridge University Press Cambridge
[28] Liebe, T.; Steinmann, P.; Benallal, A., Theoretical and computational aspects of a thermodynamically consistent framework of geometrically linear gradient damage, Comput. methods appl. mech. engrg., 190, 6555-6576, (2001) · Zbl 0991.74010
[29] Malvern, L.E., Introduction to the mechanics of continuous media, (1969), Prentice-Hall Englewood Cliffs, NJ
[30] Maugin, G.A., Internal variables and dissipative structures, J. non-equilib. thermodyn., 15, 173-192, (1990)
[31] Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity, Int. J. solids structures, 1, 417-438, (1965)
[32] Mindlin, R.D.; Eshel, N.N., On first strain-gradient theories in linear elasticity, Int. J. solids structures, 4, 109-124, (1968) · Zbl 0166.20601
[33] Mühlhaus, H.-B.; Aifantis, E.C., A variational principle for gradient plasticity, Int. J. solids structures, 28, 845-858, (1991) · Zbl 0749.73029
[34] ()
[35] Pijaudier-Cabot, G.; Bažant, Z.P., Nonlocal damage theory, J. engrg. mech., 113, 1512-1533, (1987)
[36] Polizzotto, C., Nonlocal elasticity and related variational principles, Int. J. solids structures, 38, 7359-7380, (2001) · Zbl 1014.74003
[37] Polizzotto, C., Thermodynamics and continuum fracture mechanics for nonlocal-elastic plastic materials, Eur. J. mech. A solids, 21, 85-103, (2002) · Zbl 1032.74007
[38] Polizzotto, C., Remarks on some aspects of nonlocal theories in solid mechanics, ()
[39] Polizzotto, C., 2003. Gradient elasticity and nonstandard boundary conditions. Int. J. Solids Structures, in press · Zbl 1063.74015
[40] Polizzotto, C.; Borino, G.; Fuschi, P., A thermodynamically consistent formulation of nonlocal and gradient plasticity, Mech. res. commun., 25, 75-82, (1998) · Zbl 0923.73014
[41] Polizzotto, C.; Borino, G., A thermodynamics-based formulation of gradient-dependent plasticity, Eur. J. mech. A solids, 17, 741-761, (1998) · Zbl 0937.74013
[42] Rogula, D., Introduction to nonlocal theory of material media, (), 125-222
[43] Toupin, R., Elastic materials with couple stresses, Arch. rational mech. anal., 11, 385-414, (1962) · Zbl 0112.16805
[44] Vardulakis, I.; Exadaktylos, G.; Aifantis, E.C., Gradient elasticity with surface energy: mode III crack problem, Int. J. solids structures, 33, 4531-4559, (1996) · Zbl 0919.73237
[45] Wu, C.H., Cohesive elasticity and surface phenomena, Quart. appl. math., L, 1, 73-103, (1992) · Zbl 0817.73056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.