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A Generalized Model of Oriented Continuum with Defects
Author(s) -
Stumpf H.,
Saczuk J.
Publication year - 2000
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/(sici)1521-4001(200003)80:3<147::aid-zamm147>3.0.co;2-p
Subject(s) - helmholtz free energy , constitutive equation , classical mechanics , continuum mechanics , dissipation , mechanics , physics , boundary value problem , mathematics , mathematical analysis , finite element method , quantum mechanics , thermodynamics
The aim of this paper is to present a general model of oriented continuum within the frame of Newtonian‐Eshelbian continuum mechanics to describe macro‐ and microdeformations of solids taking into account the evolution of defects as voids and cracks. Within a manifold‐theoretical setting, position‐ and direction‐dependent metric, deformation and strain measures are derived to describe macro‐ and micromotion of the body. A variational formulation is introduced leading to balance laws, boundary and transversality conditions for macro‐ and microstresses of deformational as well as configurational type, where the latter have to be satisfied by the driving forces on macro‐ and microdefects. A dissipation inequality for macro‐ and micromotion is derived via a sufficiency condition for the action integral. The Helmholtz free energy treated as the relevant thermodynamic potential is used to define thermo‐inelastic stress‐strain relations of incremental type. For the macro‐micro constitutive equations associated phenomenological macro constitutive equations are derived by introducing a second potential with corresponding evolution laws. Finally, the presented microtheory is applied to analyze the evolution of shear bands in a rod under tension and the decrease of the load‐deflection behaviour. Numerical results are given. It is shown that contrary to phenomenological theories, where shear bands are determined as bifurcation from a homogeneous state via the admittance of weak discontinuities on singular surfaces, conditions of this type are not needed within the presented micromodel of oriented continuum.