A Geometric Approach to Thermomechanics of Dissipating by Lalao Rakotomanana

By Lalao Rakotomanana

Around the centuries, the advance and development of mathematical innovations were strongly encouraged via the wishes of mechanics. Vector algebra used to be constructed to explain the equilibrium of strength structures and originated from Stevin's experiments (1548-1620). Vector research used to be then brought to review pace fields and strength fields. Classical dynamics required the differential calculus built by way of Newton (1687). however, the idea that of particle acceleration used to be the place to begin for introducing a based spacetime. on the spot speed concerned the set of particle positions in house. Vector algebra idea was once now not enough to check the various 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 proper mathematical constitution for this delivery was once 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 options have been made via Einstein in 1905 (special relativity) and 1915 (general relativity) through the use of Riemannian connection. a little bit later, nonrelativistic spacetime together with the most gains of normal relativity I It took approximately one and a part centuries for connection idea to be approved as an self reliant conception in arithmetic. significant steps for the relationship inspiration 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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The dual of the derivative of w with respect to B is equal to the total derivative of the dual of w: ~ drp* (dB w) = (drp*w). 67) Proof From the definition, the dual of the p-form w by the motion rp to the initial configuration Bo gives, for any p-plet of vectors (UIO, . . , upo) in Bo: drp*W(UIO, . , upo) = w[drp(ulO) , ... ,drp(upo)]. The derivative of the p-form w on B with respect to B is then written as: dB diw(drpulO, ... , drpupo) d dt [w (drpulO , . . , drpupo)] -d [drp *W(UIO, ... , upo)].

3 (B -derivative) Consider a continuum B with a velocity field v. 60) dt dB dt dB dt d -Wo dt dt These three definitions are compatible. A vector, I-form, and volume form are said to be instantaneously embedded in the motion of B during the lapse of time [to t + dt] if they satisfy respectively the relations: dB -u=o dt dB -w=o dt dB -wo =0. 4 (B -derivative) Let A be a mixed tensorfield ofthe type (p, q) defined on B. The time derivative of A with respect to the continuum B is a tensor of the same type as A, satisfying for any p-plet of vectors (UI, ...

It is no longer possible to find a homeomorphism to recover the initial configuration from the actual configuration. Le and Stumpf [108] have established for elastic-plastic continua the following result: The statement that the present configuration B is non-Euclidean is equivalent to the statement that plastic strain (defined as nongeometrically nonholonomic deformation) occurs in the body. The comparison of the three methods based respectively on the internal variables, the intermediate incompatible transformations (elastic and plastic), and the connections geometry developed here is certainly an extensive task but it should be clarified in the future.

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