Integration of the constitutive equation

Size: px

Start display at page:

Download "Integration of the constitutive equation"

Ethel Stewart
5 years ago
Views:

1 Inegaion of he consiive eqaion REMAINDER ON NUMERICAL INTEGRATION Analyical inegaion f ( x( ), x ( )) x x x f () Exac/close-fom solion (no always possible) Nmeical inegaion. i. N T i N [, T ] [ i, i ] i [ i, i ] Time ineval/sep "i" Δ i i Time incemen x( ) x g( x, x, x,...) Appoximae solion (epening on i i i i i he vales a pevios iscee imes)

2 Inegaion of he evolion eqaion Loaing/Unloaing eqaions Time inegaion λ ; ; (, ) g λg λ g λg g τ ( ) f ( ) a) ELASTIC STATE ï ì g ºe - < e () No evolion of ïì < í l í ïî e () < ïî e () ³ b) INELASTIC (DAMAGE) STATE b-) UNLOADING ì ï g ºe - í ïî e () g º () - < l No evolion of ( ) < e b-) LOADING g º () - e b--) NEUTRAL LOADING l No evolion of () b--) PURE LOADING l ( ) > Evolion of () ³ e e e e

3 Inegaion of he evolion an consiive eqaions SUMMARY : ) he iniial vale of is o > ) he iniial vale of is ( ) 3) neve eceases ( ³ ) e 4) is always smalle o eqal han (g º - ) e e if e gows e ³ 5) When e eqals hen ìï í ïî ï if e eceases e < ; e () max(, τ ( s)) s [,] τ s hisoical máximm of [, ( )] () s q q( ) s τ q ( ) : The inegaion is exac (close fom). No epenence on D, τ

4 Tangen consiive opeao (efiniion) (,) x (,) x Σ((,)) x Tangen consiive opeao Σ( ) ( ) ang () Σ( ij ) ij( ) [ ang ] ijkl i, j, k, l {,,3} kl kl Tangen consiive opeao Secan consiive opeao

5 Tangen consiive opeao (analyical eivaion) Tangen consiive moli ((), ) ( ( )) ( ): ( ()) ang () : ( ) ( ) ang () ( ) : ang : ang (): Deivaion ( ) : : : ang Elasic egime/unloaing/neal loaing ( ) : ( ) ( ang ang sec secan consiive opeao)

6 Tangen consiive opeao (analyical eivaion) Inelasic egime-loaing q( ) ( ) τ () q H H () 3 τ > q '( ) q( ) () '( ) τ : : : : ( : : ) τ : : : : : : : : : : : ; : : : q H () 3 q H : : 3 : : τ

7 Tangen consiive opeao (analyical eivaion) H q ( ) : : ( ) : ; : : 3 q H q H ( ) : ( : ) : [( ) ( [ : ])]: 3 3 ang : q H ang () ( ) ( [ : ]) 3 Symmeic moel (ension/compession) ang : : : : ( ) (elasic/nloaing) () q H ( ) ( ) (loaing) 3 ang

8 Inegaion of he consiive eqaion: Ienificaion of he cen sae Ienificaion a he en of he cen ime ineval) ) Elasic gδ τ Δ Δ < τ Δ < Δ Δ Δ Δ ) Unloaing/Neal loaing τ Δ [, Δ] gδ τ Δ Δ τ τ Δ Δ Δ Δ Δ Δ 3) Loaing gδ τ Δ Δ Δ > τ Δ τ > Δ Δ > Δ > Δ Δ > Δ <

9 Inegaion of he consiive eqaion Nmeical algoihm: INPUT DATA [, Δ],, Sep Sep ( Compe If τ Δ s Δ Δ τ ( ) s Δ q( Δ) Δ Δ Δ Δ ang ) ( ) Δ Δ Δ Δ : Δ : : : s Δ Δ Δ Δ Elasic Unloaing Neal loaing EXIT

10 Inegaion of he consiive eqaion Nmeical algoihm: Sep 3 Δ Δ If τ > τ Δ Δ q( Δ) Δ ( ) Δ Δ Δ (Loaing) q( ) H ( ) ( ) ( ) Δ Δ Δ ang Δ Δ 3 Δ Δ ( Δ ) EXIT OUTPUT DATA [, Δ],,( ) Δ Δ ang Δ

11 Chaaceizaion of he Haening/Sofening law Linea haening/sofening H ( ) [, ] ( ( q ) q( ) q( ) H q > [, ) q( ) H [, ) H [, )

12 Chaaceizaion of he Haening/Sofening law Exponenial haening/sofening A ( ) q q q e A ( ) ( ) ( > ) q( ) H ( ) A e ( q ) A ( )

13 Chaaceizaion of he Haening/Sofening law Vale of I is compe fom he D case niaxial elasic limi τ : : : E ( ) : ij ij E : E

14 Chaaceizaion of he elasic omain { f τ q } Elasic omain Ε : (, ) ( ). Symmeic (ension/compession) moel τ q τ : : ( ) τ ( ) : : τ : : 3 E Remak Eqal ension an compession elasic limis.

15 Chaaceizaion of he elasic omain. Tensile-amage-only moel Posiive conepa of a scala fncion (McAley backe) x x > x x < Posiive conepa of a sess enso In maix foma: [ ] iagonalizaion [ ] iag 3 en o oiginal [ ] iag sysem of [ ] Posiive 3 cooinaes x conepa of [ ] x

16 Chaaceizaion of he elasic omain In enso foma: i 3 i eigenveco"" i i 3 ˆ ˆ i pi pi i eigenvale"" i pˆ pˆ i i i an have he same eigenvecos shaes he posiive eigenvales of has hose negaive eigenvales of nll τ τ ( ) ( ) Sain nom eefiniion τ : : ( ) : : ( ) : : ( ) τ ( : : ) :

17 Chaaceizaion of he elasic omain a) If b) If > > 3 > 3 3 Pe ensile sae : : τ τ Pe compessive sae < < τ : : f τ < 3 < 3 The sae is always elasic Inelasic ν ν E ν ν Elasic

18 Chaaceizaion of he elasic omain Non-symmeic ension-compession moel θ τ θ : : n > θ τ 3 3 θ,, 3 : : i i,, 3 < θ τ : : 3 3 n i i n E n n

19 . Viscoamage moel Ch.. Coninm Damage Moels

20 Moel chaaceizaion The ae effecs can be accommoae ino he invisci amage moel (ae inepenen) Evolion eqaion (,),, [, ] λ Consiive eqaion ( ) : Damage fncion g(, ) τ Kash-Khn Tcke an pesisency coniions λ, g λg if g λg

21 Moel chaaceizaion Visco eglaizaion (Pezyna s Reglaizaion) λ g η Obaine by eplacing in he invisci moel Evolion eqaion λ(,) g η Consiive eqaion whee η is he viscosiy The only change is he evolion law is he McAley backe ( ) : Damage fncion g(,) τ Kash Khn Tcke an pesisency coniions No necessay x x> x x < x x > x x x < x x x

22 Moel chaaceizaion Remak ; g η λ g η λg g g á gñ ( λη) ηλ η η η η g Remak g λη η As λη g g Remak 3 λ g λg λg g λ g If η λ g As η λ g λg If g λ g Loaing/nloaing coniions Pesisency coniions As η (invisci case ) he ae inepenen coninm amage moel is ecovee

23 Moel chaaceizaion Remak 3 g τ > τ > Fo η an λ hen g λη f τ q > τ > q The sess/sain sae can lay osie he elasic omain Δ 3 Δ 3 τ q τ 3 Remak 4 f g g η f > g > g > η 3 No evolion ( nloaing / neal loaing) Evolion ( loaing)

24 Fee enegy an issipaion Fee Enegy ψ (, ) ( ) ψ ( ) ( ) ( : : ) Dissipaion : ψ Coleman's Theoem ψ(, ()) ψ(, ()) : : ψ (, ()) ( ( )) ψ ( ()), ( ( )) : ψ ( ()), : ψ :

25 Tangen consiive enso Tangen consiive opeao (,) (()) Invisci moel ( as a paamee) ((),) Viscos moel ( as an inepenen vaiable) ( ) : : ( ) : ( ( )) : '( ) ( ( )) : '( ) η g ( ( )) : '( ) g(, ) η (,) : ( ) '( ) g(, ) : η v ang

26 Time epenency/rae epenency Remak ((),) The sess enso can change even if emains consan Invisci Viscos

27 END OF LECTURE 4

Orthotropic Materials

Orthotropic Materials Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε