Dear Agrawal, what puzzles me in you question is your reference to the "loading in one direction". Now, the response to a monodimensional loading is not bound to be monodimensional (think e.g. of squeezing a toothpaste tube). Thus, no matter what your free energy function is like I am afraid you have to invoke the same old textbook formula which is valid for the general case (see e.g. Landau and Lifschitz' Theory of elasticity): the amount of dissipated energy per unit time and volume is just the scalar obtained by the inner product of the stress tensor with the deformation tensor, with full contraction of all indices. 

Dear Andrea Di Vita, Thank you for your response. I have considered multidimensional response to the loading and have calculated the total amount of energy in a loop time by the same old textbook formula that you have stated. Here, my question was  "Is there any other way around through which I can calculate the same thing using thermodynamic variables ?" Although I have tried to derive them using first and second law of thermodynamics but at certain point I  needed function of entropy and I was not able to proceed further. I have attached a file in which all these things are in detail.Please look in section 2 of attached file.

Thanking you,

Dear Agrawal, what do you mean by 'pressure' in the Gibbs' free energy of the system you are dealing with? Indeed, this is the hydrostatic pressure in homogeneous and isotropic fluids and gases. As for thermodynamics, in the general case pressure is just equal to minus partial derivative of Helmolz' free energy on volume at constant temperature; this definition allows explicit computation of pressure from the partition energy of an arbitrary system in thermodynamical equilibrium. As for mechanics, the diffrential of mechanical work you are referring to with the quantity -pdV  is again a linear functional of the components of the deformation tensor. Here you are skipping the whole question just because you assume all components of T but T11 to vanish. In the general case, however, thermodynamics alone cannot say what the dissipation in your system is like, precisely because thermodynamics provides you with general information which hold regardless of the detailed nature of the dissipative processes in your system. 

Dear Andrea Di Vita, I am not very sure about the pressure but my explanation is as viscoelastic materials are in between viscous and elastic materials. So, the pressure should corresponds(or related) to viscous part of deformation which is same in starting and in ending of loop. That's why I have taken \int dP as zero.Finally, thank you for answering my question that with present knowledge Thermodynamics alone cannot calculate dissipation energy.  It would be great if you can refer some text that contains idea about pressure in solids/viscoelastic materials to me.

Thanking you,

Dear Agrawal, unfortunately I am no expert. My knowledge is at the level of the basic notions discussed e.g. in  https://en.wikipedia.org/wiki/Viscoplasticity

Apart from some standard textbook on solid state phsics (Kittel), it comes to my mind that Mie-Gruneisen law, i.e. a standard form for the equation of state in solids, is discussed in 

https://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state

Dear Andrea Di Vita, I am writing this to close this question and to let you know that we can calculate dissipation energy by using different form of Second law of Thermodynamics that is, Clausius- Duhem Inequality or Dissipation Inequality.

Thanking you,

Entropy: A concept that is not a physical quantity

https://www.researchgate.net/publication/230554936_Entropy_A_concept_that_is_not_a_physical_quantity