Elastoplasticity problems 169 and finally △eH(x)=△Eu&)+△Eu(x) (3.44) The following notation is applied (3.42)and (3.43): Bx)=p{(&) (3.45) B2s(区)=p&p(xg (3.46) B(x)=xi(x) (3.47) All these equations are substituted into the variational formulation of the problem(cf.(3.39)).There holds 3Chn△Eu△emn=Chmm△Eu+△Eu)△Em+△Enm） =Cklmn (AEgAEmn+△Eu△E,n+△Eu△Emm+AEgAEmn） (3.48) =号Chn⑧i△nB点An”+略AnB△△ +B5△△BA”+BfA△n△A） 6u△lu△4u=6up(K△upi(xAu (3.49) Next,the following notation is applied: kge=∫6up(ug9i(aa2 (3.50) 2 kg=∫CumB Bd2 (3.51) 42 k号=∫分Cmn®2+2S+2Ba)dn (3.52) 2 where 缨=kg+k+k号 (3.53) and for the second and third order termsElastoplasticity problems 169 and finally ( ) x ( ) x ( ) x kl kl kl ∆ε = ∆ε + ∆ε (3.44) The following notation is applied (3.42) and (3.43): () () x x ζ ζ Bkl ϕk ,l (1) = (3.45) () () () ( ) , , (2) N kl i k i l B uξ ζ ζ ξ x = ϕ x ϕ x (3.46) ( ) x () () x x ζξ ζ ξ Bkl 2 ϕi,k ϕi,l 1 = (3.47) All these equations are substituted into the variational formulation of the problem (cf. (3.39)). There holds ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 1 2 1 2 1 N N mn N N kl N mn N N kl N N mn N kl N mn N klmn kl klmn kl mn kl mn kl mn kl mn klmn kl mn klmn kl kl mn mn B u u B u B u u B u u C B u B u B u B u u C C C µ ν µν ζ ζ ζξ µ µ ζ ξ ζξ µ ν µν ζ ζ ξ ξ ζ ζ ε ε ε ε ε ε ε ε ε ε ε ε ε ε + ∆ ∆ ∆ + ∆ ∆ ∆ ∆ = ∆ ∆ + ∆ ∆ ∆ = ∆ ∆ + ∆ ∆ + ∆ ∆ + ∆ ∆ ∆ ∆ = ∆ + ∆ ∆ + ∆ (3.48) () () ( ) , ( ) 2 , 1 2 , , 1 ~ ~ N i l N kl i k i l kl i k u u u uξ ξ ζ ζ σ ∆ ∆ = σ ϕ x ∆ ϕ x ∆ (3.49) Next, the following notation is applied: () () ∫ Ω = Ω e k u d i l N kl i k e x x ξ ζ σ ζ ζξ σ ϕ ϕ , ( ) , ( ) ~ (3.50) ∫ Ω = Ω e k CklmnBkl Bmn d con e ζ ξ ζξ (1) (1) 2 ( ) 1 (3.51) ∫ ( ) Ω = + + Ω e k C B B B B B B d klmn kl mn kl mn kl mn u e ζ ξ ζ ξ ζ ξ ζξ (1) (2) (2) (1) (2) (2) 2 ( ) 1 (3.52) where e e con e u e k k k k (1) ( ) ( ) ( ) ζξ ζξ σ ζξ = ζξ + + (3.53) and for the second and third order terms