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Introduction to Mechanics of Continua 25
Using then the equality v - [T]n = [T]v -n, a consequence of the defi-
nition of the transposed tensor and of the symmetry of the stress tensor,
precisely
v-(T)jn =[T]’v-n=([T]v-n_ [Appendix Al,
the first integral of the right side, f v - [T]nda, becomes
Ss
[¥-(tnda = [ aivttlvay = [ (00 dv.
S D D
Since f pf-vdv = f (pau; — 743,70;) dv, from the Cauchy equations,
D D
taking into account that the second order tensor [G] = grad v (of compo-
nents v;,;) can be split as a sum of a symmetric tensor [D] of components
Diy = § (vig +05) (the rate-of-strain tensor) and a skew-symmetric
tensor [Q] of components Q;; = 4 (vi,; — v;,4) (the rotation tensor) while
f [G]-[T]dv = f [D] - [T]dv, we finally have
D D
éL _ D1 2 _ -
a= / 2
[T]av + 55 | ov dvu=W+ Ec,
D D
where W is the internal(deformation) energy whose existence is cor-
related with the quality of our continuum to be deformable (for rigid
bodies obviously W = O) while Ee is the kinetic energy of the system.
Usually a specific deformation energy w is defined by 2w = [T] - [D] =
tr ((T][D]) and then W = 2 [ wdv.
D
Part of the work done, contained in W, may be recoverable but the
remainder is the Jost work, which is destroyed or dissipated as heat due
to the internal friction.
So we have, in the language of deformable continua, the result of en-
ergy conservation which states that the work done by the forces exerted
on the material subsystem P is equal to the rate of change of kinetic
energy Ec and of internal energy W.
2.7 General Conservation Principle
The integral form of mass conservation, momentum torsor and energy
principle as established in the previous section respectively, can all be
joined together into a unique general conservation principle. Precisely,
for any material subsystem P C M, which occupies the configuration
D Cc D whose boundary is S, at any moments ¢; and t2, we have the
following common form for these principles:
