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.

I have a question that has troubled me for quite a while. Any kind of help would be greatly appreciated. I am computing the strain-energy density in a elastic solid when wave passes through. The formula I use is U=[I1^2+2(1+v)I2]/(2E), where U is the strain-energy density, I1 and I2 the first and second principal stress invariants, v poisson's ratio=0.3, E the Young's modulus. Since I1^2 must be positive, I2 might be negative behind the wave front. U might be negative. However, the textbook and many papers I ran into, says strain-energy density must be positive.