## Abstract

The basic roots and meaning of conservation laws are reminded in Appendix B in relation with the concept of the invariance group. The scope of this book is rather limited to classical thermomechanics laws. The basic problem in deriving the conservation laws focuses on calculating the evolution of any *p*-form field during a continuum deformation. For the sake of simplicity, consider a continuum *B* of dimension three, with an affine connection ∇, a metric **g**, and a volume form ω, in motion with respect to a Galilean reference. The motion is defined by the application φ_{t-t0} the velocity field of which is transcribed as **v**(*M*, *t*).

## Keywords

Volume Form Helmholtz Free Energy Boundary Action Exterior Derivative Integral Invariance
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## References

- 1.The usual version of Cauchy’s theorem starts with a scalar-valued flux. Let there be three quantities
*pe, r*_{e}, and*J*_{e}respectively of class*C*^{1}*,C*^{0}, and C^{1}verifying the conservation laws. Assume that the flux*J*_{e}is a function of the unit normal vector**n**at ∂*B: J*_{e}=*J*_{e}**(n)**. Then a unique vector field**J**_{e}exists such that (this vector is called the “flux density vector”)*J*_{e}(**n**) =**J**_{e}.**n**.Google Scholar - 2.Cauchy’s theorem (3.12) implies that there exists a vector field
**J**_{C}(contact force intensity) such that ω_{C}=*h*_{0}(J_{C}) and consequently*dh*_{0}(J_{C}) = div J_{C}ω_{0}. Moreover, it can be shown that J_{C}= v · σ. The exterior derivative of the mechanical action density is written: \( d\omega _C = dh_0 (p_n^i ) \otimes e_i = div (p_n^i )\omega _0 \otimes e_i . \) Google Scholar

## Copyright information

© Springer Science+Business Media New York 2004