By Lalao Rakotomanana
Across the centuries, the advance and development of mathematical strategies were strongly encouraged through the desires of mechanics. Vector algebra used to be built to explain the equilibrium of strength structures and originated from Stevin's experiments (1548-1620). Vector research used to be then brought to review speed fields and strength fields. Classical dynamics required the differential calculus constructed by way of Newton (1687). however, the idea that of particle acceleration was once the place to begin for introducing a based spacetime. instant pace concerned the set of particle positions in house. Vector algebra conception used to be no longer enough to match the several velocities of a particle during time. there has been a necessity to (parallel) shipping those velocities at a unmarried element earlier than any vector algebraic operation. definitely the right mathematical constitution for this delivery used to be the relationship. I The Euclidean connection derived from the metric tensor of the referential physique used to be the single connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime recommendations have been made via Einstein in 1905 (special relativity) and 1915 (general relativity) by utilizing Riemannian connection. a bit of later, nonrelativistic spacetime such as the most positive factors of basic relativity I It took approximately one and a part centuries for connection thought to be authorized as an self sustaining thought in arithmetic. significant steps for the relationship proposal 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
Elastic finite strain. Let us consider a continuum B undergoing finite deformation. The transfonnation of a vector Uo E TMo Bo (of the tangent space of the continuum) gives the vector u E TM B: u du dt dcp(uo) arp = ax (uo) a acp --(uo) at ax a arp av av = --(uo) = -(uo) = -(u). (V'v - V'vT) and the strain rate tensor D == ! (V'v+ V'vT). In elastic finite strain theory it is usual, although not necessary, to define the Green-Lagrange strain measure: c(uoa , UOb) == I 2 [g(Ua, Ub) - g(UOa, UOb)].
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). at . 70) Let w be a I-form uniform and constant field on the continuum B , meaning that ~~ = O. 70), we obtain d -d u(M, t) t a = -u(M , t) + Vv(M t)u(M , t). at . 71) are the classical formulae of Euler, extended to continuum with singularity, by using non (sym)-metric connection. Let 1/1 = w(u) now be a scalar field defined on B where wand u are respectively form and vector fields.
The internal energy e(M, t) or, alternatively, the Helmholtz free energy l/J(M, t), 6. the heat flux vector q(M, t), 7. the body force pb(M, t), 8. the volume heat source pr(M, t). , , . The constitutive variables 1 and 2 are the primal variables and from 3 to 6 the dual variables. Body forces and volume heat source are of secondary interest in continuum thermomechanics. In the present section we deal with the formulation of contact mechanical action and contact heat action at the continuum boundary, From the axiomatic point of view, the existence of a stress tensor a and a heat flux vector Jq is not postulated a priori.