# Geometry and Kinematics

• L. R. Rakotomanana
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 31)

## Abstract

We fix once and for all on a referential solid body endowed with a metric tensor g and a positive volume form ω0. The reference is assumed Euclidean, that is, characterized by the existence of a global Cartesian coordinate system covering the whole referential body. The vector space underlying the reference is denoted ∑.

## Keywords

Vector Field Volume Form Torsion Tensor Actual Configuration Affine Connection
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## References

1. 1.
A continuum is also assumed to be a measured space, thus endowed with a non-negative scalar measure called the mass distribution of the continuum $$m(B) = \int_B \rho (M,t)dv$$. The volume element dv indicates that the integral is defined in terms of Lebesgue measure in Euclidean space. Integration on a continuum, as a manifold, is introduced later.Google Scholar
2. 2.
The continuum B, endowed with a metric tensor g that is symmetric positive definite (and a volume form ω0 supposed positive), induced by the Euclidean referential body, is a Riemannian manifold (and oriented). A fundamental theorem of differential geometry may then be applied at each deformation state: Let B be a Riemannian continuum (class C2), then there exists one and only one affine connection ∇ on B (usually called Riemannian connection) such that ℵ = 0 and ℜ = 0. The proof is classical in differential geometry. Starting from a null torsion and from the covariant derivative of the metric tensor, the explicit expression of this Riemannian connection is directly obtained [29]: $$m(B) = \int_B \rho (M,t)dv$$.Google Scholar
3. 3.
Consider a continuum B and a vector field u on B. Previous studies, e.g., [149] defined the equations of motion for such a vector field as: $$\bar \Gamma _{ab}^c = \frac{1} {2}g^{cd} [u_b (g_{ad} ) + u_a (g_{bd} ) - u_d (g_{ab} )] - \frac{1} {2}(\aleph _{0ba}^c + g^{cd} g_{ae} \aleph _{0bd}^e + g^{cd} g_{eb} \aleph _{0ad}^e ).$$ where A(u) represents the slippage of the vector u with respect to the continuum (the second-order tensor A is called the slip tensor). Comparison with relation (2.58) shows that the derivative with respect to the continuum also characterizes the slippage of the vector u with respect to the matter. The “slippage” theory has been used to study the constitutive laws of polymeric fluids consisting in flexible macromolecules, rodlike molecule s suspended and interacting with the fluid, e.g., [105].Google Scholar