Cosserat Theories: Shells, Rods and Points by M.B. Rubin

By M.B. Rubin

Continuum Continuum mechanics mechanics presents presents a a theoretical theoretical constitution constitution for for examining interpreting the the reaction reaction of of fabrics fabrics to to mechanical mechanical and and thermal thermal so much. quite a bit. One considered one 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, strength power and and entropy) entropy) are are legitimate legitimate for for all all basic basic fabrics. fabrics. so much so much of of the the trendy glossy study learn in in continuum continuum mechanics mechanics focuses specializes in at the the advance improvement of of constitutive constitutive equations equations which that are are used used to to represent signify the the reaction reaction of of a a specific specific classification category 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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Extra info for Cosserat Theories: Shells, Rods and Points

Example text

15) does not require the distance between these two points to remain constant in time (which would correspond only to a rigid motion of the continuum). 15) requires the changing distance between the two points to remain the same in both the present and the superposed configurations. 9) which requires the length of the spring to remain the same in the simple example discussed previously. 16) where e(t) is an arbitrary vector function of time only and Q(t) is an arbitrary orthogonal second order tensor function of time only QQT =QTQ=I .

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). Therefore, if more than three independent kinematic constraints are imposed, then f* will have more than three independent components which are arbitrary functions of position and time. This means that even when appropriate boundary conditions are specified, it will not be possible to uniquely determine all components of f*.

K g k' . ] g kn , [. 2) of the Christoffel symbols [mi,k] of the first kind and r mi~ of the second kind. Moreover, it follows that V'i = = [vkli ] gk ' vkli = = v'i • gk . 5) - ki. 6) vk of the vector v is where the covariant partial derivative vkli of the covariant components vk defined by v kli = vk'i - vm r ki m . 5) for the gradient of a tensor, it follows that the gradient of v with respect to x * can be expressed in the forms grad* v = v'i ® gi = [v [vkli] kli ] (gk®gi) = [v\ ] (gk®gi) .

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