Μηχανική Συνεχούς Μέσου
Continuum Mechanics, Μηχανική Συνεχών Μέσων
For a more general derivation using tensors, we consider a moving body (see Figure), assumed as a continuum, occupying a volume V, at a time t, having a surface area S, with defined traction or surface forces per unit area represented by the stress vector acting on every point of every body surface (external and internal), body forces Fi per unit of volume on every point within the volume V, and a velocity field vi, prescribed throughout the body. Following the previous equation, the linear momentum of the system is:
By definition the stress vector is defined as , then
Using the Gauss's divergence theorem to convert a surface integral to a volume integral gives (we denote as the differential operator):
Now we only need to take care of the right side of the equation. We have to be careful, since we cannot just take the differential operator under the integral. This is because while the motion of the continuum body is taking place (the body is not necessarily solid), the volume we are integrating on can change with time too. So the above integral will be:
Performing the differentiation in the first part, and applying the divergence theorem on the second part we obtain:
Now the second term inside the integral is: Plugging this into the previous equation, and rearranging the terms, we get:
We can easily recognize the two integral terms in the above equation. The first integral contains the convective derivative of the velocity vector, and the second integral contains the change and flow of mass in time. Now lets assume that there are no sinks and sources in the system, that is mass is conserved, so this term is zero. Hence we obtain:
putting this back into the original equation:
For an arbitrary volume the integrand itself must be zero, and we have the Cauchy's equation of motion
As we see the only extra assumption we made is that the system doesn't contain any mass sources or sinks, which means that mass is conserved. So this equation is valid for the motion of any continuum, even for that of fluids. If we are examining elastic continua only then the second term of the convective derivative operator can be neglected, and we are left with the usual time derivative, of the velocity field.
If a system is in equilibrium, the change in momentum with respect to time is equal to 0, as there is no acceleration
or using tensors,
These are the equilibrium equations which are used in solid mechanics for solving problems of linear elasticity. In engineering notation, the equilibrium equations are expressed in Cartesian coordinates as
- E. Becker und W. Bürger, Kontinuumsmechanik, Teubner, 1975
- R. L. Bisplinghoff, J. W. Mar and T. H. H. Pian, Statics of Deformable Solids, Dover, 1965.
- P. Chadwick, Continuum Mechanics, Dover, 1976.
- W.F. Chen and D. J. Han, Plasticity for Structural Engineers, Springer, 1988.
- D.C. Kay, Tensor Calculus, Schaum’s Outline Series,1988
- L.E. Malvern, Introduction to the Mechanics of Continuous Medium, Prentice-Hall, 1969.
- Sommerfeld, Mechanik der deformierbaren Medien, Bd. II, Verlag Herri Deutsch, 1992.
- A.J.M. Spencer, Continuum Mechanics, Dover, 1980
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