Mori_Tanaka effective Stiffiness/Compliance Formula for inclusions with similar shape

when all the inclusions with similar shape are embeded into the matrix, the 
 Mori_Tanaka effective Stiffiness/Compliance Tensor can be , respectively, given by

L ={∑c_r(L₀^*+L_r)^-1}^-1-L₀^*

M={∑c_r(M₀^*+M_r)^-1}^-1-M₀^* 

where,

L₀^*=L₀(S₀^-1-I),

M₀^*=[(I-S₀) ^-1-I]M₀

In my opinion, similarity in shape means all the inclusions has the same Eshelby Tensor.

Then from the Mori Tanaka formula

 L =∑^N_r=0 c_rL_rT_r (∑^N_n=0 c_nT_n)^-1

T_r =[I+S_rL₀^-1(L_r-L₀)]^-1

=[I+S_oL₀^-1(L_r-L₀)]^-1 

= [I-S_o+S_oL₀^-1L_r]^-1

= [S_o(S_o^-1-I)+S_oL₀^-1L_r]^-1

= [S_oL_o^-1L_o(S_o^-1-I)+S_oL₀^-1L_r]^-1

= {S_oL_o^-1[L_o(S_o^-1-I)+L_r]}^-1

= {S_oL_o^-1[L_o^*+L_r]}^-1

=[L_o^*+L_r]^-1L_o S_o^-1

so

 L =∑^N_r=0 c_rL_rT_r (∑^N_n=0 c_nT_n)^-1

={∑^N_r=0 c_rL_r[L_o^*+L_r]^-1}L_o S_o^-1{[∑^N_n=0 c_n[L_o^*+L_n]^-1] L_o S_o^-1 }^-1

={∑^N_r=0 c_rL_r[L_o^*+L_r]^-1}L_o S_o^-1S_oL_o ^-1{∑^N_n=0 c_n[L_o^*+L_n]^-1}^-1

 ={∑^N_r=0 c_rL_r[L_o^*+L_r]^-1} {∑^N_n=0 c_n[L_o^*+L_n]^-1}^-1

Is the procedure right?

and how to make the following derivation?

Can anyone help me?