O viskoplastičnim polikristalima prividno inkrementalnog tipa (nastavak)

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1 O viskoplastičnim polikristalima prividno inkrementalnog tipa (nastavak) Milan V. Mićunović Prezentacija na Mašinskom fakultetu u Beogradu DIVK, 17. februar 2011.

2 2. Deo zasnovan na radu u Phi. Mag (ref. [7]) 2

3 Part A: THEORY A1. Geometric issues towards evolution equations RVE plastic distortion ( free meso rotation ) micro plastic distortion ( constrained micro rotation ) residual micro elastic stretch Π : 1 A micro plastic distortion ( only by slips) P T 1 T CP C( Π ) Π Π Π Π V V macroplastic deformation tensor by means of microplastic deformation tensors for individual grains F P C 1/2 P DΠ A D DA P 3

4 DR ΩR constrained micro spin RVE(t) F E (t) B(t) T 0 A ( t) R ( t) A ( t ) R ( t) F P (t)= U P (t) F(t) T DA Ω A A Ω RVE(t 0 ) F E (t 0 ) B (t 0 ) DCP DC( Π) R P =1 [Zorawski,1972] Micro macro evolution equations 4

Neale &Toth & Jonas (J2-model) - reference [7] d d / / 0 0 0 m1 Rate-independent model (insensitive on initial yield stress Y) 0 my0 ( initial shear

5 Neale &Toth & Jonas (J2-model) - reference [7] d d / / m1 Rate-independent model (insensitive on initial yield stress Y) 0 my0 ( initial shear yield stress m orientation factor) Micunovic & Albertini & Montagnani model reference [6] d exp( M ) sign( ) ( Y ) deq Y 0 1 b ln( Deq ) Y fast medium slow 5

6 A2. Evolution equation of quasi-rate independent materials Tensor representation D P D eq J c S Y c S 1 d 0 1 d d Y 2 0 Accumulated plastic strain Triggering kernel (slide 5) t ( t) D ( t') dt ' J( t t') D ( ') ' 0 0 eq t dt 0, t' t* J ( t t ') exp M, t ' t* 1 2 h A B Important note for loading function 1 2 normality D P S does not hold t 6

7 Part B: Numerical experiments on slightly disordered random grain distributed RVE-s 7

8 Numerical procedure of integration of field equations Choice of constitutive model NTJ-model or MAM-model 3 g s g s N 5 grains N 12 slip systems : {1, N }, {1, N } 0 R ( random grain rotations ) A 0 s n ht () 3 2 t max t Stress history T (0)11 (0)12 T 0 0 max 0 tension,...,t shear ,, T( t) T h( t) Y t 10,10,10 s local T local T R TR T : A 8

9 Π ( t t) Π ( t) d A da P P T P C( Π ) Π Π C C( Π ) Here 1/ 2 P P P P Re laxed Taylor ' s assumption T E E E U Π Π C U F R 1 1/ 2 Π volume aveaging ( M. Zorawski) 1 E UPΠP Micro rotation 1 E E R Π U Micro spin Ω R ( t t) R ( t) / t T da ΩA A Ω 9

10 Number of active slip systems Low speed loading NTJ (J2-model) Quasi-rate independent 10

11 Number of active slip systems Quasi-rate independent model MAM Medium speed High speed 11

12 Number of active slip systems Neale & Toth & Jonas (J2-model) + artificial (Dg,Y)-sensitivity Medium speed High speed 12

13 Active slip systems distribution at low speed Quasi-rate independent model 13

14 Active slip systems distribution at low speed Neale & Toth & Jonas (J2-model) 14

15 Active slip systems distribution at medium speed Quasi-rate independent model 15

16 Active slip systems distribution at medium speed Neale & Toth & Jonas (J2-model) 16

17 Active slip systems distribution at high speed Quasi-rate independent model 17

18 Macroscopic and average residual stresses for xz-shear and zz-unitension at low speed (~0.001/s) Quasi-rate independent model 18

19 Macroscopic and average residual stresses for xz-shear and zz-unitension at medium speed (~1/s) Quasi-rate independent model 19

20 Concluding remarks Evolution equation formed by tensor representation having incremental form is postulated to model inelastic metals. The rate dependence takes place by means of stress rate dependent value of the initial yield stress. This approach for auste-nitic stainless steels has permitted exceptionally good agreement with dynamic experiments performed at JRC-Ispra, Italy at strain rates in the interval [0.001, 1000] 1/s. This theory is applied to slightly disordered fcc-polycrystals. For some characteristic given stress histories (leading to low, medium and high strain rates) number of active slip systems and average Taylor's m-number for RVE with 1000 grains are found and compared with so-called J2-approach. Possible practical applications could be: Calculation of grain interactions where grains have crystals of diverse orientations (Hungarian 3 X 3 X 3 cube) by FEM Calculation of residual stresses in heat affected zone and welded materials composed of grains of diverse size 20

21 References [1] Hill R, J. Mech. Phys.Solids, 1965,Vol.13/4, p.213 [2] Kroener E, J. Eng. Mech. Division, 1980, p [3] Levin V M, Izv. ANSSSR Mekhanika Tverdogo Tela, 1976, No.6, p.137 [4] Maruszewski B, Micunovic M, Int.J.EngineeringScience, 1989, No.27, p.955. [5] Micunovic M., In: STRUCTURED MEDIA (Kroener s memory symp., Poznan 2001 [6] Micunovic M, Philosophical Magazine, 2005, Vol. 85, Nos , p 4031 [7] Neale KW, Toth LS, Jonas JJ, Int.J.Plast.,6, (1990), p45. [8] Rice J R, J. Mech. Phys. Solids, 1971, Vol. 19, p.433. [9] Vakulenko A A, Izv.AN SSSR Mekhanika Tverdogo Tela, 1970, No.1, p.69. [10] Zorawski M, private communication, [11] Kudrjavceva L, Micunovic M, Proceedings of YUMECH-2001, Belgrade,

THERMOMECHANICS OF SOFT INELASTICS BODIES WITH APPLICATION TO ASPHALT BEHAVIOR

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