Elio Sacco. Dipartimento di Ingegneria Civile e Meccanica Università di Cassino e LM

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1 Elio Sacco Diartimento di Ingegneria Civile e Meccanica Università di Cassino e LM

2 1D erect elastolastic resonse OA elastic resonse AB nonlinear resonse, constant stress BC elastic nloading CB elastic reloading B - nonlinear resonse, constant stress y A Rheological model y E B ε, ε e ε ε = ε + ε e elastic art o the total strain, recovered strain lastic art o the total strain, irreversible strain O ε C ε e ε

3 yield nction ( ) = 0 0 OA elastic resonse AB nonlinear resonse, constant stress BC elastic nloading CB elastic reloading B - nonlinear resonse, constant stress ( ) < 0 ( ) = 0 ( ) 0 ( ) 0 < < ( ) = 0 ε = ε > ε = ε = ε > A Rheological model y E B ε, loading- nloading K conditions 0 ε 0 ε = 0 consistency condition = 0 O ε C ε e ε 3

4 otal strain elastic strain lastic strain Constittive law lastic evoltion: loading-nloading K condition consistency ε= ε + ε ε e ε e e ( ) = Cε = C ε ε = λ 0 λ 0 λ = 0 = 0 normality rle lastic mltilier 4

5 Constittive laws Yield nction with (isotroic) hardening: evoltion law loading-nloading K consistency = 0 0= + q q = C( ) H γ = C λ H λ ( ) = Cε ε q= Hγ ( ) ( ), q = eq 0 + q = = = = eq q 0 λ 0 λ = 0 eq λ λ γ λ λ λ C C + H = 5

6 Constittive laws Yield nction with (kinematic) hardening: evoltion law loading-nloading K consistency = 0 0= X ( β ) = ( ) C H X ( ) = C λ λh X X X = C ε ε β= Hε X= β ( ) 0 X = Xeq X = λ X X 0 λ 0 λ = 0 λ X eq = X C ( C+ H) eq X 6

7 Drcker stability ostlate 1) Let the material sbjected to an admissible stress state : ( ) 0. ) hen, let an increment o the stress be alied sch that = +, with () 0. 3) Finally, the stress increment is removed. According to Drcker: the material is stable i the work erormed dring the closed stress cycle is not negative ater removing the stress increment. 0 * * O ε ( ) < 0 ( ) = 0 7

8 Conseqences o the Drcker stability ostlate 1) Let : 0 and : 0, it reslts ) Let : < 0 and : = 0, it reslts ( * ) * ( * ) > 0 : < 0 ( * ) * ( * ) 0 : 0 3) he yield locs has to be convex 1 1 : 0 : 0 ( ) ( ) 4) Normality rle: the lastic strain rate is orthogonal to the tangent lane in : = 0, ( ) 1 = α + β, α + β = 1 : 0 * ε ε 8

9 Conseqences o the Drcker stability ostlate 5) Let = 0 < 0, it reslts = φ ( ) 0 with φφ the dissiated ower er nit volme. 6) Let the incremental stress corresonding to the incremental lastic strain εε, it is: 0 7) I the material is erectly lastic, it is: 0 = 9

10 Static and kinematic theorems o (erect) lasticity b body orces srace orces Ω Ω µ collase loading mltilier stress ield in corresonce o collase strain and dislacement ields in corresonce o collase φ = dissiated ower er nit volme ( ) ( ) PLV ( ) ( ) ( ) Ω ( ε ) φ Ω ε = µ b+ µ = b+ Ω Ω Ω Ω ( ) ( ) Ω Ω Ω 10

11 Remark 1 Dring the condition o inciient collase, it is 1 µ ( ) + ( ) = ( ) = ( ) + ( e) = ( ) + ( ) b C Ω Ω Ω Ω Ω Ω Ω i.e., in incremental orm 1 µ ( ) + ( ) = ( ) + ( ) b C Ω Ω Ω Ω Dring the collase, no increment o loads is ossible, then µ = 0 he last term is always ositive as C is ositive deined, then = 0 As conseqence e = 0 = Remark Dring the condition o inciient collase, the work erormed by the orces is non negative: µ ( ) + ( ) = ( ) 0 ( ) + ( ) 0 b b Ω Ω Ω Ω Ω 11

12 Static theorem µ collase loading mltilier stress ield in corresonce o collase strain and dislacement ields in corresonce o collase ε s µ loading mltilier s stress ield ε b PLV ( ) = µ ( ) + ( ) Ω Ω Ω s s ( ) = µ ( ) + ( ) Ω Ω Ω eqilibrated s µ s b, µ s, s admissible ( ) 0 ε b s s ( ) ( ) = ( µ µ ) ( ) b+ ( ) Ω Ω Ω 1

13 PLV ε b s s ( ) ( ) = ( µ µ ) ( ) + ( ) Ω Ω Ω becase o the Drcker stability ostlate Ω ( ) ( s ) 0 ε then, it reslts: s µ µ 13

14 Kinematic theorem µ collase loading mltilier stress ield in corresonce o collase k k k, ( 0) e = k µ loading mltilier k kinematically admissible dislacement and strain ields sch that stress ield ε b k k k PLV ( ) = µ ( ) + ( ) Ω Ω Ω ε b k k k k k ( ) ( ) = ( µ µ ) ( ) b+ ( ) Ω Ω Ω k k k k k ( ) = µ ( ) + ( ) Ω Ω Ω L ( ) b ( ) 0 = + > k k k Ω Ω 14

15 PLV k k k k k ( ) ( ) = ( µ µ ) ( ) + ( ) ε b Ω Ω Ω he Drcker stability ostlate can be rewritten as: Ω ( k ) ( k ) 0 ε then, it reslts: k µ µ Finally s k µ µ µ 15

16 M maximm admissible bending moment Static theorem: deine a stress state eqilibrated and admissible s q µ = M 8 s µ = 8M q 16

17 M maximm admissible bending moment Kinematic theorem: deine a lastic kinematic ield M M / / k µ q ϕ M ϕ 3M ϕ = 0 4 ϕ k µ = 1 M q 17

18 Mises (native) lane stress case, q = eq 0 + q eqivalent stress nd 1 invariant o the deviatoric stress J = Q= eq = J s Qs deviatoric stress s = = s= M M = Derivatives o the eq J s = eq J yield nction s 1 = MQM eq 1 J J s = 1 = = = Qs M eq J J s [ ] ( ) ( ) ( ) q = 1 18

19 rial soltion = γ = γ n ( ) tr tr = Cε n q = n ( 0 q ) = + tr tr tr eq Hγ n i tr < 0 the trial is the soltion o the roblem else a correction is reqired: constittive eqations evoltion consistency = Cε n q = H γn + γ 1 = λ = λ MQM J γ = λ = λ q ( ) ( ) ( q) = MQM 0 + the set o eqations can be redced to ( ) R = Cε = 0 n 1 R = λ MQM = 0 J R ( ) = MQM 0 + H γn + λ = 0 3 eqations 3 eqations 1 eqation 19