How to do such integration with dummy indicials for strain energy density?

Submitted by pengfei_guo on Sun, 10/13/2013 - 01:12

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 :).