By Lalao Rakotomanana
Across the centuries, the advance and development of mathematical options were strongly motivated via the wishes 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 was once then brought to check speed fields and strength fields. Classical dynamics required the differential calculus built via Newton (1687). however, the concept that of particle acceleration was once the place to begin for introducing a dependent spacetime. instant pace concerned the set of particle positions in area. Vector algebra conception used to be now not adequate to check different velocities of a particle during time. there has been a necessity to (parallel) delivery those velocities at a unmarried element ahead of any vector algebraic operation. the fitting 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 suggestions have been made through Einstein in 1905 (special relativity) and 1915 (general relativity) through the use of Riemannian connection. a little later, nonrelativistic spacetime including the most positive factors of common relativity I It took approximately one and a part centuries for connection thought to be authorised as an self sustaining concept in arithmetic. significant steps for the relationship suggestion 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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Extra resources for A Geometric Approach to Thermomechanics of Dissipating Continua
For this purpose, we consider first the Euclidean vector space E underlying the referential body. 1 (Vector-valued p-form) A vector-valued p-form field is a p-form field w at the point M of the continuum B (of finite dimension) with values in the vector space E and is thus a totally antisymmetric p-linear map from TM B into E: w: (UI, ... , up) ~ W(UI, ... , up). Consider now a base of E denoted (ei) . We can write W(UI,"" up) = Wi(UI,"" up)ei where wi (UI, ... , up) are real numbers. In some sense, we have to introduce the projection of the form on a I-form according to wi.
49) ~~ The derivation of these two sets of identities is reported in Appendix D. , , . In this sense, Edelen and Lagoudas  called the Cartan structural equations and their exterior derivatives, respectively, kinematics equations of singularity and continuity equations of singularity. The usefulness of the Bianchi identities should not be overestimated. , . 49) is that any defect state with discontinuity of scalar and vector fields that satisfies these relations is admissible. 1 (Continuum motion) Given a referential body, a motion of a continuum B, with continuous distribution of singularity, is a i-parameter family of configurations indexed by the time t.
149] defined the equations of motion for such a vector field as: du dt = V'v(u) + A(u) 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). 58) shows that the derivative with respect to the continuum also characterizes the slippage of the vector u with respect to the matter. , . 5 Kinematics of continuum 33 In this last equation, we can proceed as follows to obtain: = = d:tg(U;,Uj) = gG:(U;),Uj)+g(Ui , ::(Uj)).
A Geometric Approach to Thermomechanics of Dissipating Continua by Lalao Rakotomanana