On nonlocal gradient model of inelastic heterogeneous media

Stumpf, H.; Saczuk, J

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On nonlocal gradient model of inelastic heterogeneous media

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Stumpf H. , Saczuk J

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Nielokalny model gradientowy niesprężystych ośrodków heterogenicznych

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Abstrakty

The aim of this paper is to investigate the influence of nonlocality on the physical and material field equations of heterogeneous media. Taking into account that plastic deformations in metals or damage in brittle and ductile materials are governed by physical mechanisms observed on levels with different lengthscales, we introduce a 6-dimensional kinematical concept with two locally defined vectors to model the material behaviour on a macro- and meso- or microlevel. Using a variational procedure the physical and material balance laws, boundary and transversality conditions are derived for macrp- and microdeformations of heterogeneous media. The disspation inquality including relaxation terms for transport processes is presented. The constitutive equations are formulated with macro- and microstrain measures, their gradients and time rates, and the anisotropy tensor as arguments, where the latter can be considered as a coupling measure between the deformed macrostates with compatible microstates. The model presented in this paper delivers a framework, which enables one to derive various nonlocal and gradient theories by introducing simplifying assumptions. As the special case a solid-void model is considered.

Celem pracy jest zbadanie wpływu nielokalności na fizyczne i materialne równania pola ośrodków heterogenicznych. Biorąc pod uwagę, że plastyczna deformacja w metalach lub zniszczenie w kruchych i ciągliwych materiałach rządzone są przez fizyczne mechanizmy na różnych poziomach skali, wprowadzono 6-wymiarową strukturę z dwoma lokalnie zdefiniowanymi wektorami do modelowania materialnego zachowania ośrodka na poziomie makro- i mezo- lub mikroskali. Wykorzystując wariacyjną procedurę otrzymano fizyczne i materialne prawa bilansu, warunki brzegowe i transwersalność dla makro- i mikrodeformacji ośrodków heterogenicznych. Przedstawiona nierówność dyssypacyjna zawiera człony relaksacyjne procesów transportu. Sformułowane równania konstytutywne wyrażono przy pomocy miar makro- i mikroodkształcenia, ich gradientów i przyrostów oraz tensora anizotropii, gdzie ostatni argument może być traktowany jako miara sprzężenia pomiędzy odkształconymi makrostanami i kompatybilnymi mikrostanami. Przedstawiony w pracy model dostarcza podstaw, które poprzez wprowadzenie uproszczających założeń umożliwiają otrzymanie różnych postaci nielokalnych i gradientowych teorii. Jako przypadek szczególny rozpatrzono model typu ciało stałe-pustka.

Słowa kluczowe

microstructure nonlocal inelasticity gradient theory configurational forces

Wydawca

Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej

Czasopismo

Journal of Theoretical and Applied Mechanics

Rocznik

2002

Tom

Vol. 40 nr 1

Strony

205--234

Opis fizyczny

Bibliogr. 65 poz., rys.

Twórcy

autor

Stumpf H.

Lehrstuhl fur Allgemeine Mechanik, Ruhr-Universitat Bochum, Bochum, Germany

autor

Saczuk J

jsa@imp.gda.pl

Institułe of Fluid-Flow Machinery, Polish Academy of Sciences, Gdańsk

Bibliografia

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47. PAGANO S., ALART P., MAISONNEUVE O., 1998, Solid-solid phase transition. Local and global minimization of non-convex and related potentials. Isothermal case for shape memory alloys, Int. J. Engng Sci., 36, 1143-1172
48. PEIERLS R.E., 1940, The size of a dislocation, Proc. Phys. Soc. Lond., 52, 14
49. RAKOTOMAMANA R., 2001. Connecting mesoscopic and macroscopic scale leng-ths for ultrasonic wave characterization of micro-cracked materiał, Math. Mech. Solids, (in the review procedurę)
50. RiCE J.R., 1968, A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35, 379-386
51. ROEHL D., RAMM E., 1996, Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept, Int. J. Solids Struct., 33, 3215-3237
52. ROGULA D., 1973, On nonlocal continuum theories of elasticity, Arch. Mech., 25, 233-251
53. RUND H., 1966, The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand, London
54. SACZUK J., STUMPF H., VALLEE, C, 2001, A Continuum model accounting for defect and mass densities in solids with inelastic materiał behaviour, Int. J. Solids Struct, (in print)
55. SCHIECK B., SMOLEŃSKI W.M., STUMPF H., 1999, A shell finite element for large strain elastoplasticity with anisotropies. Part I: Shell theory and variational principle. Part II: Constitutive eąuations and numerical applications, Int. J. Solids Struci, 36, 5399-5424; 5425-5451
56. SIENIUTYCZ S., 1981a, Thermodynamics of coupled heat, mass and momentum transport with finite wave speed. I - Basic ideas of theory, Int. J. Heat Mass Transfer, 24, 1723-1732
57. SIENIUTYCZ S., 1981b, Thermodynamics of coupled heat, mass and momentum transport with finite wave speed. II - Examples of transformations of fluxes and forces, Int. J. Heat Mass Transfer, 24, 1759-1769
58. STOUT R.B., 1981, Modelling the deformations and thermodynamics for materials involving a dislocation kinetics, Crystal Lattice Defects, 9, 65-91
59. STUMPF H., SACZUK J., 2000, A generalized model of oriented continuum with defects, Z. Angew. Math. Mech., 80, 147-169
60. STUMPF H., SACZUK J., 2001, On a generał concept for the analysis of crack growth and materiał damage, Int. J. Plasticity, 17, 991-1028
61. TOUPIN R.A., 1962, Elastic materials with couple-stresses, Arch. Rat. Mech. Anal, 11, 385-414
44. NiLSSON C, 1998, On nonlocal rate-independent plasticity, Int. J. Plasticity, 14, 551-575
45. NOWACKI W., 1986, Theory of Asymmetńc Elasticity, Pergamon Press, Oxford/PWN, Warsaw
46. PAG ANO S., ALART P., 1999, Solid-solid phase transition modelling: relaxation procedures, configurational energies and thermomechanical behaviours, Int. J. Engng Sci, 37, 1821-1840
47. PAGANO S., ALART P., MAISONNEUVE O., 1998, Solid-solid phase transition. Local and global minimization of non-convex and related potentials. Isothermal case for shape memory alloys, Int. J. Engng Sci., 36, 1143-1172
48. PEIERLS R.E., 1940, The size of a dislocation, Proc. Phys. Soc. Lond., 52, 14
49. RAKOTOMAMANA R., 2001. Connecting mesoscopic and macroscopic scalę leng-ths for ultrasonic wave characterization of micro-cracked materiał, Math. Mech. Solids, (in the review procedurę)
50. RiCE J.R., 1968, A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35, 379-386
51. ROEHL D., RAMM E., 1996, Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept, Int. J. Solids Struct., 33, 3215-3237
52. ROGULA D., 1973, On nonlocal continuum theories of elasticity, Arch. Mech., 25, 233-251
53. RUND H., 1966, The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand, London
54. SACZUK J., STUMPF H., VALLEE, C, 2001, A Continuum model accounting for defect and mass densities in solids with inelastic materiał behaviour, Int. J. Solids Struct, (in print)
55. SCHIECK B., SMOLEŃSKI W.M., STUMPF H., 1999, A shell finite element for large strain elastoplasticity with anisotropies. Part I: Shell theory and variatio-nal principle. Part II: Constitutive eąuations and numerical applications, Int. J. Solids Struci, 36, 5399-5424; 5425-5451
56. SIENIUTYCZ S., 1981a, Thermodynamics of coupled heat, mass and momentum transport with finite wave speed. I - Basic ideas of theory, Int. J. Heat Mass Transfer, 24, 1723-1732
57. SIENIUTYCZ S., 1981b, Thermodynamics of coupled heat, mass and momentum transport with finite wave speed. II - Examples of transformations of fluxes and forces, Int. J. Heat Mass Transfer, 24, 1759-1769
58. STOUT R.B., 1981, Modelling the deformations and thermodynamics for ma-terials involving a dislocation kinetics, Crystal Lattice Defects, 9, 65-91
59. STUMPF H., SACZUK J., 2000, A generalized model of oriented continuum with defects, Z. Angew. Math. Mech., 80, 147-169
60. STUMPF H., SACZUK J., 2001, On a generał concept for the analysis of crack growth and materiał damage, Int. J. Plasticity, 17, 991-1028
61. TOUPIN R.A., 1962, Elastic materials with couple-stresses, Arch. Rat. Mech. Anal, 11, 385-414
44. NiLSSON C, 1998, On nonlocal rate-independent plasticity, Int. J. Plasticity, 14, 551-575
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46. PAG ANO S., ALART P., 1999, Solid-solid phase transition modelling: relaxation procedures, configurational energies and thermomechanical behaviours, Int. J. Engng Sci, 37, 1821-1840
47. PAGANO S., ALART P., MAISONNEUVE O., 1998, Solid-solid phase transition. Local and global minimization of non-convex and related potentials. Isothermal case for shape memory alloys, Int. J. Engng Sci., 36, 1143-1172
48. PEIERLS R.E., 1940, The size of a dislocation, Proc. Phys. Soc. Lond., 52, 14
49. RAKOTOMAMANA R., 2001. Connecting mesoscopic and macroscopic scalę leng-ths for ultrasonic wave characterization of micro-cracked materiał, Math. Mech. Solids, (in the review procedurę)
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51. ROEHL D., RAMM E., 1996, Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept, Int. J. Solids Struct., 33, 3215-3237
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54. SACZUK J., STUMPF H., VALLEE, C, 2001, A Continuum model accounting for defect and mass densities in solids with inelastic materiał behaviour, Int. J. Solids Struct, (in print)
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59. STUMPF H., SACZUK J., 2000, A generalized model of oriented continuum with defects, Z. Angew. Math. Mech., 80, 147-169
60. STUMPF H., SACZUK J., 2001, On a generał concept for the analysis of crack growth and materiał damage, Int. J. Plasticity, 17, 991-1028
61. TOUPIN R.A., 1962, Elastic materials with couple-stresses, Arch. Rat. Mech. Anal, 11, 385-414
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