By Lalao Rakotomanana
Across the centuries, the advance and progress of mathematical suggestions were strongly encouraged by way of the desires of mechanics. Vector algebra used to be built to explain the equilibrium of strength platforms and originated from Stevin's experiments (1548-1620). Vector research was once then brought to review speed fields and strength fields. Classical dynamics required the differential calculus constructed by means of Newton (1687). however, the idea that of particle acceleration used to be the start line for introducing a based spacetime. immediate pace concerned the set of particle positions in house. Vector algebra idea was once no longer adequate to match the several velocities of a particle during time. there has been a necessity to (parallel) delivery those velocities at a unmarried element prior to any vector algebraic operation. 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 one connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime recommendations have been made by means of Einstein in 1905 (special relativity) and 1915 (general relativity) through the use of Riemannian connection. a bit of later, nonrelativistic spacetime together with the most positive aspects of basic relativity I It took approximately one and a part centuries for connection idea to be accredited as an self sufficient concept in arithmetic. significant steps for the relationship idea are attributed to a sequence of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.
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Extra info for A Geometric Approach to Thermomechanics of Dissipating Continua
Ax} ax' C dx C dX'fi S . = 0 dX'dX} in which, C = as, we have applied the Stokes theorem. Since the curve C is arbitrary, the integrand also vanishes. If the triplet (f1, f2 , f3) is a basis of the tangent space, then there exists a set of real numbers (called coefficients of the connection) such that rt We deduce that the tensor field, called the torsion tensor, ~ = (r~j - rJi )fi ® f j ®fk vanishes on the continuum B. Explicitly, it is straightforward to calculate the coefficients of the connection as = uifi on B in which the components u i are assumed of class C 2 .
5 (Dissipation principle) For all members ofthe class of thermomechanical processes that are admissible for an internal constitutive assumption, the rate of entropy production must have nonnegative values. Remark. , .
We thus have: d a (Ub)] -[u dt dua = -(Ub) +U a (dUb) . dt dt Since Ub is also an embedded vector, we can easily deduce: dUb dt dua dt This second relation allows us to check the compatibility of the above three definitions on the embedded vector, form, and volume form. 5 (Velocity ofcontinuum) Consider a continuum B undergoing holomic and nonholomic deformations, characterized by the displacement field u(M, t) and the connection V (M, t) . For the sake ofcovariance and frame indifference. kinematics is described by the metric geM , t) .
A Geometric Approach to Thermomechanics of Dissipating Continua by Lalao Rakotomanana