I don't think, the expressions i) and ii) yield a meaning. The elastic energy density \xi for a general stress state is given by the double dot product of the stress and strain tensors, \sigma and \epsilon,

\xi = 1/2 * \sigma_ij \epsilon_ij (using Einstein's summation convention).

With Hooke's law for a general stress-strain-state

\sigma_ij = C_ijkl \epsilon_kl,

or written with the forth order compliance tensor D,

\epsilon_ij = D_ijkl \sigma_kl

the energy density as a function of the stress tensor is

\xi = 1/2*\sigma_ij \sigma_kl D_ijkl.

The expressions \xi = \sigma_0^2/(2E) and \xi = \tau^2/(2G) correspond to the special stress states of uniaxial tension

\sigma_xx = \sigma_0, \sigma_ij = 0 else

and pure uniaxial shear

\sigma_xy = \sigma_yx = \tau, \sigma_ij = 0 else.

The expressions i) and ii) though, don't correspond to any simple state.

Consult the following

https://books.google.co.uk/books?id=1dUKAAAAQBAJ&pg=PA319&lpg=PA319&dq=(%CF%842)+/+(2G)&source=bl&ots=5RfbdX1Hrw&sig=MbJoLdz3jkPXbeBHsOXaPhoVQZI&hl=en&sa=X&ved=0ahUKEwiTyb-jucjWAhWDL8AKHclxAugQ6AEINzAD#v=onepage&q=(%CF%842)%20%2F%20(2G)&f=false

This depends on the stress states.  You can calculate energy density for each mode seperately

I thank all those who are academically excellent, had respected my technical query, and spent their valuable time in writing detail expressions with the supporting documents and text books.

Question asked were:

What could be the form of energy density calculated by the expression

i) (τ^2) / (2E); ii) (σ^2) / (2G)

Now the first form of energy in question is explained.

I have named it as Bond Energy; {(τ^2) / (2E)}V.

Bond energy, is the first mechanical energy term of 21st century.

I had developed a closed-form mathematical solution and proposed a novel failure criterion called as Bond failure criterion that estimates the ultimate failure load of adhesively bonded joints in CFRP composite structures: For aerospace application.

The ultimate failure load estimated by Bond failure criterion is compared with the mechanically tested failure load measured from the third party test specimens, and found in agreement within 1-11% difference.

Description of third party test specimens validated by Bond Energy Method:

1) Single side patch repaired bonded joint (differed by 5.42%)

2) Double side patch repaired bonded joint (differed by 1.28%)

3) Circular step patch repaired bonded joint (differed by 2.02%)

4) 2 Step patch repaired bonded joint (differed by 9.5%)

5) 4 Step patch repaired bonded joint (differed by 11.1%)

6) 8 Step patch repaired bonded joint (differed by 6.19%)

Reference:

Polagangu James et al . Estimation of Failure Load of Composite Bonded Joints by Using 1D 2D FE Analysis and Mathematical Equation of Strain Energy in Adhesive Layer. International Conference on Composite Materials and Structures, ICCMS 2017, Paper ID 257, 27-29 Dec 2017, Indian Institute of Technology Hyderabad, Telangana state. India.

Third party test specimens validated from the following references:

[1] Mohammad Kashfuddoja., M, Ramji., 2013. Whole-field strain analysis and damage assessment of adhesively bonded patch repaired CFRP laminates using 3D-DIC and FEA. Composite. Part B, 53, 46-61.

[2] Kumari Asha, Byji Varughese, D, Saji., 2013. Stress analysis of stepped lap bonded repair joint under compressive loading. Proceedings of ISAMPE National Conference on Composites INCCOM-12, CP160. Bangalore, India.

[3] Zong-Hong Xie., Xiang Li., Sui-An Wang., 2016. Parametrical study on stepped-lap repair of composite laminates. The 2016 Structures Congress. Structures16. Jeju Island, Korea.