On the geometric character of stress in continuum mechanics

Abstract

Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000).

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

“…More detailed discussions of the associated differential geometry can be found in [1,24,31]. Differential forms arise as a natural approach in formulating conservation laws and relations in continuum mechanics [10,16]. The exterior calculus provides a convenient means to generalize the Stokes Theorem, Divergence Theorem, and Green's Identities to curved spaces [20].…”

“…Application areas include hydrodynamics within fluid interfaces [5,28], electrodynamics [32], stable methods for finite elements [3,4,8], and geometric processing in computer graphics [7,11,21,34]. The exterior calculus of differential geometry provides a cooordinate invariant way to formulate equations on manifolds with close connections to topological and geometric structures inherent in mechanics [16,20]. The exterior calculus provides less coordinate-centric expressions for analysis and numerical approximation that often can be interpreted more readily in terms of the geometry than alternative approaches such as the tensor calculus [10,16,35].…”

“…The exterior calculus of differential geometry provides a cooordinate invariant way to formulate equations on manifolds with close connections to topological and geometric structures inherent in mechanics [16,20]. The exterior calculus provides less coordinate-centric expressions for analysis and numerical approximation that often can be interpreted more readily in terms of the geometry than alternative approaches such as the tensor calculus [10,16,35]. Many discrete exterior calculus approaches have been developed for efficient low-order approximations on triangulated meshes [11,13,21].…”

Discrete differential forms for computational modeling

“…For applications based on this exterior calculus or other geometric algebras, see [4,14,3,10,18,35]. The reader interested in the application of differential forms to E&M is further referred to [38], for applications in fluid mechanics see [28], and in elasticity see [25] and [16]. The reader is also invited to check out current developments of variants of DEC, for instance, in [9,36,40,20].…”

On Spatial Covariance, Second Law of Thermodynamics and Configurational Forces in Continua

“…Moreover, he derived the standard splitting of the total energy into internal and kinetic energies and the transformation law for external forces. Since then a lot of effort has been placed and a series of papers have appeared in the literature dealing with the concept of covariant energy balance in both a conservative (see, e.g., Yavari et al [10]; Kanso et al [11]; Yavari and Ozakin [12]; Yavari and Marsden [13]; Panoskaltsis and Soldatos [14]) as well as a dissipative (see, e.g., Yavari [15]; Panoskaltsis et al [16,17]) setting.…”