By Lalao Rakotomanana
Across the centuries, the improvement and progress of mathematical innovations were strongly motivated via the wishes of mechanics. Vector algebra used to be constructed to explain the equilibrium of strength platforms and originated from Stevin's experiments (1548-1620). Vector research was once then brought to check speed fields and strength fields. Classical dynamics required the differential calculus constructed by way of Newton (1687). however, the concept that of particle acceleration was once the start line for introducing a based spacetime. instant speed concerned the set of particle positions in house. Vector algebra concept was once no longer adequate to match different velocities of a particle during time. there has been a necessity to (parallel) delivery those velocities at a unmarried aspect earlier than any vector algebraic operation. definitely the right mathematical constitution for this shipping was once the relationship. I The Euclidean connection derived from the metric tensor of the referential physique was once the one connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime options have been made through Einstein in 1905 (special relativity) and 1915 (general relativity) through the use of Riemannian connection. just a little later, nonrelativistic spacetime consisting of the most gains of common relativity I It took approximately one and a part centuries for connection thought to be accredited as an self sustaining thought in arithmetic. significant steps for the relationship inspiration are attributed to a chain of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.
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Additional resources for A Geometric Approach to Thermomechanics of Dissipating Continua
15) Remarks. 1. 8), it is clear that the resultant boundary actions JaBWe and JaBwH do not necessarily define the stress vector p~ and the heat flux Jq uniquely. Indeed, we may add either to the stress tensor (J or to the heat flux vector Jq any tensor fields ij and i q such that div ij = 0 and div i q = O. In other words, the problem of unicity should not be overlooked. Gurtin and Williams have studied the problem in depth and proposed a theory that admitted continuity assumptions such as the existence of a positive number s.
4 Spatial and material accelerations Consider a body B the velocity field of which is denoted v. The total variation of a scalar field 1/I(M, t) on B between two instants t and t + dt is (total variation means change due to both time and position): . I d1/l(M , t) = lIm -[1/I(M e..... O e + evdt, t + edt) -1/I(M, t)] . 69) 36 2. Geometry and Kinematics A straightforward calculationh shows that the total derivative of 1/1 is, V denoting the affine connection on the continuum: d -d 1/I(M, t) t a = -1/I(M , t) + Vv(M t)1/I(M, t).
2 (Velocity field) Consider a continuum B that deforms with respect to a referential body L with the displacement field u(M, t) . Let (el, e2, e3) be any vector base field embedded in L. The velocity field of B with respect to L is defined as the vector field v(M, t) d . == -[u(M, t)(e')]ei. 54) dt The motion rp(M, t) can be also used instead of the displacement field to define the velocity. " The definition is merely an extension of the very classical definition of particle velocity in rigid body kinematics.