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How to do such integration with dummy indicials for strain energy density?
The integration puzzles me is
Integrate C_ijkl e_kl de_ij (from 0 to e_ij )= 1/2 C_ijkl e_ij e_kl
which is the well known strain energy density of elasticity and much like the energy in a spring as
Integrate k x dx (from 0 to x ) = 1/2 k x x.
By the way, this integration appears at page 226 in the book "Nonlinear Finite Elements for Continua and Structures" by T Belytschko, B Moran, WK Liu, 1999, the book just directly throws out the result without any details.
So my questions are:
1. How to derive this tensor integration?
2. How to do such tensor integrations in a general way for more complicated integrands with more dummy and free indicies ?
Thanks.
Sorry for my poor english, and I am still improving it :).
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