Elastoplasticity problems 171 a=Ae+2ga (3.59) ao）n,=ce）n,and）n=c.(ae）a (3.60) Cem =C +C (3.61) Further,using spatial averaging definitions,the averaged stress tensor components are calculated as follows: (e,)o=8 and (o)o=o (3.62) Hence,the effective elasticity tensor components C are derived for a given increment as do=C de (3.63) c-立cc+cb:。 (3.64) In the particular case of a two-component composite,the transformation and concentration matrices are obtained as,cf.(3.30)and(3.31) D=I-A)(C-C2)C (3.65) D21=(I-A2)C1-C2)PC (3.66) D2=-(I-A)C1-C2)-C3 (3.67) D2=-I-A2)C-C2)'C2 (3.68) C.C2 denote here the components corresponding to elastic properties,while A,,A,are mechanical concentration matrices.Finally,using (3.64)it is obtained that ci-立GC+c,Cg+sDe':o+e,De'o (3.69) +cD:d+Dd: The FEM aspects of TFA computational implementation are discussed in detail in Section 3.4 below.Further,it should be noticed that there were some approaches in the elastoplastic approach to composites where,analogously to the linearElastoplasticity problems 171 ∑ = = + N A D r,s 1 r s r d d d Ω Ω ε ε εinel r rs s (3.59) r r el r el r d d Ω Ω σ C ε el = r and r r inel r inel r d d Ω Ω σ C ε = r (3.60) r el C = Cr + C eff r (3.61) Further, using spatial averaging definitions, the averaged stress tensor components are calculated as follows: ε ε Ω r = and σ σ Ω r = (3.62) Hence, the effective elasticity tensor components eff C are derived for a given increment as σ = dε eff d C (3.63) ( ) inel r N r,s inel rs N r el r r eff C c C D ε : σ 1 1 r s 1 c − = = = ∑ + ∑ (3.64) In the particular case of a two-component composite, the transformation and concentration matrices are obtained as, cf. (3.30) and (3.31) 1 1 11 1 1 2 D (I A )(C C ) C− = − − (3.65) 1 1 21 2 1 2 D (I A )(C C ) C− = − − (3.66) 2 1 12 1 1 2 D (I A )(C C ) C− = − − − (3.67) 2 1 22 2 1 2 D (I A )(C C ) C− = − − − (3.68) 1 2 C ,C denote here the components corresponding to elastic properties, while 1 2 A ,A are mechanical concentration matrices. Finally, using (3.64) it is obtained that ∑ ∑ () () = − − = + + + N r eff el el inel inel inel inel c c 2 2 1 1 2 12 2 1 1 1 2 2 1 11 1 C C C c D ε :σ c D ε :σ ∑ () () − − + + inel inel inel inel 2 1 1 2 22 2 1 1 21 1 c D ε :σ c D ε :σ (3.69) The FEM aspects of TFA computational implementation are discussed in detail in Section 3.4 below. Further, it should be noticed that there were some approaches in the elastoplastic approach to composites where, analogously to the linear