By M.B. Rubin
Continuum Continuum mechanics mechanics offers offers a a theoretical theoretical constitution constitution for for studying reading the the reaction reaction of of fabrics fabrics to to mechanical mechanical and and thermal thermal lots. quite a bit. One one in every of of the the beauties beauties of of continuum continuum mechanics mechanics is is that that the the elemental basic stability stability legislation legislation (conservation (conservation of of mass mass and and balances balances of of linear linear momentum, momentum, angular angular momentum, momentum, power strength and and entropy) entropy) are are legitimate legitimate for for all all basic basic fabrics. fabrics. such a lot so much of of the the trendy glossy study study in in continuum continuum mechanics mechanics focuses makes a speciality of at the the advance improvement of of constitutive constitutive equations equations which that are are used used to to represent represent the the reaction reaction of of a a selected specific type type of of fabrics fabrics (e.g. (e.g. invisicid invisicid fluids, fluids, viscous viscous fluids, fluids, elastic elastic solids, solids, viscoelastic viscoelastic solids, solids, elastic elastic plastic solids, elastic-viscoplastic solids, etc.). plastic solids, elastic-viscoplastic solids, etc.).
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Of the stress T "'". 32) of the balance momentum. 8) is properly invariant under SRBM. (v) The tensors f* and y* are both independent of the rate L*. 5), can be rewritten in the form (f* - g-112 y* y*). y*) 0 n* = 0 . 11) vanishes. 5) for arbitrary values of the other components of n*. 13) where y* remains an arbitrary function of (x *,t) that is determined by the equations of motion and the boundary conditions. 14) In general, the constraint response f* influences the equation of linear momentum but not the conservation of mass or angular momentum equations (since it is a symmetric tensor).
9) For clarity in some instances, the value of a function and its specific functional form are distinguished by using an additional symbol over the usual symbol used for the value of the function. Specifically, an overbar ( ) is used when the independent space variable is /\ 1\ -Y, a hat ( ) is used when the independent space variable is X *, and a tilde ( ) is used when the independent space variable is x *. Moreover, f(Y,t) is called the material representation, f(X*,t) is called the Lagrangian or referential representation, and f(x*,t) is called the Eulerian or spatial representation.
8) J* > 0 . ~*(X*,t) X*(Y) i*(y,t) Fig. 1 Material description and reference and present configurations Since the above mappings are invertible, a general scalar or tensor function f can be expressed in terms of three possible representations - 1\ * . , * f = f(Y,t) = f(X ,t) = f(x· f(x' ,t) . 9) For clarity in some instances, the value of a function and its specific functional form are distinguished by using an additional symbol over the usual symbol used for the value of the function. Specifically, an overbar ( ) is used when the independent space variable is /\ 1\ -Y, a hat ( ) is used when the independent space variable is X *, and a tilde ( ) is used when the independent space variable is x *.
Cosserat Theories: Shells, Rods and Points by M.B. Rubin