Geometry of finite deformations and time-incremental analysis

An Intrinsic Geometric Formulation of Hyper-Elasticity, Pressure Potential and Non-Holonomic Constraints

“…Remark 5.2. The related questions of how to recast locally such a boundary condition on the body B, in a so-called intrinsic manner [42,49,25,20], and of how to redefine a first Piola-Kirchhoff tensor on B are answered in Appendix E.…”

“…Intrinsic formulations nowadays use modern tools in differential geometry [35,32,15,33,55]. In this direction, we insist on the fundamental role played by the manifold of all the Riemannian metrics on the body -introduced in solids mechanics by Rougée [49,51,52] and Fiala [18,19,20] -in the formulation of hyper-elasticity (see section 3).…”

“…Formulating the problem in an intrinsic manner, meaning on the body, was initiated by Rougée [51,52], who defined an elasticity law as a vector field on the manifold of Riemannian metrics on the body. However, the formulation of the boundary conditions on B rather than on a reference configuration in ambient space seems to remain an open problem in the general case [54,25,20].…”

Objective rates as covariant derivatives on the manifold of Riemannian metrics

“…The geometrization of Continuum Mechanics was continued afterwards, under the impulse of Marsden and Hughes [48], and is still alive today [14,72,69,78,77,25,76,36,60,10,18,5,75,70]. The possibility to make mechanical calculations (and numerical discretizations in [19,20]) directly on the body B emerges with the formulation, by Noll [57] and later by Rougée [63], of so-called intrinsic stresses, and by the possibility to recast boundary conditions on the abstract manifold B (see Noll's formulation in [59]). In line with Eringen [15], Green and Zerna [28], Benzecri [3], Noll [57,59], and then Epstein and Segev [14], Rougée [63,65,66,67] has furthermore rightly understood the fundamental role played in Continuum Mechanics by the manifold of Riemannian metrics on the body B.…”

An intrinsic geometric formulation of Hyper-elasticity, pressure potential and non-holonomic constraints

“…In modern continuum mechanics formulation, these configurations are described using mappings from some abstract manifold with boundary, the body B, into the ambient (usually Euclidean) space E [52,51]. From the pure differential geometry point of view, the body does not have to be embedded in space [34,41] and the formulation of continuum mechanics, more precisely of the boundary conditions, directly on the body boundary ∂B seems to remain an open question in the general case [22,17]. Note that Noll [34,35] did introduce the denomination "intrinsic" as synonymous of "defined on the body B".…”