a mechanism for ductile fracture in shear viggo...

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A MECHANISM FOR DUCTILE FRACTURE IN SHEAR

Viggo TvergaardDepartment of Mechanical Engineering,

Solid MechanicsSolid MechanicsTechnical University of Denmark, Bldg. 404,

DK-2800 Kgs. Lyngby, Denmark

DTU Mechanical Engineering, Solid Mechanics, Technical University of Denmark

N.A. Fleck, J.W. Hutchinson & V. Tvergaard, JMPS (1989)


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N.A. Fleck, J.W. Hutchinson & V. Tvergaard, JMPS (1989)


V. Tvergaard, Int. J. Mech. Sci. (2008)

Fig. 1. Periodic array of cylindrical voids used to model ductile failure in shear.


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conditions of simple shear

/ij ijG g

2J flow theory

1/ 2ij ij ijG g ij ijkkL

1/

/ ,

/ / ,

YN

Y Y Y

E

E

Mises stress1/2(3 / 2)ij

e ijs s

1 k , , , ,12

kij i j j i i k ju u u u

, ,ij ij k i ij i

ij i k j i ij iV A V Au u dV T u dA dV T u dA


1 2 20, forI IIu U u U x B


IU IIUis prescribed, and is calculated such that the stress ratio on the top surface has the prescribed value

22 12/

0 0

2 1 21 1for

A A

T dx T dx x B 0 0

22 1 12 1 00 0

, for2 2A A

T dx T dx x BA A

periodicity conditions 1 1 2 2

1 2 1 2,u u u u

1 1 2 21 2 1 2,T T T T

/ 0.002 ,Y E 0.3 0.1N 0 0/ 4B A

average shear angle , defined by

tan /IU B


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1 2 2 2 20( ) ( ) ,x x R 1 2 0 .T T The void surface, is initially stress free so that

At a later stage of the deformation a hydrostatic pressure p is applied inside the voids to simulate the effect of crack surfacecontact in a relatively simple manner. The nominal tractions and their increments on the void surface are

,i ir i ir irr rT p n T p p n

, ,

1

2ir ijk mr

j j km k mg u g u

: length of the void when it deforms into an ellipsoidal cross-section

:vV void volume per unit length in the direction3x

/ :vw V the average width of the void

Then, the average aspect ratio of the void is required to satisfy the inequality

/w

When there is no particle inside, the void will collapse completely and then the internal pressure could approximately describe frictionless sliding of the opposite void surfaces against each other.



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.



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IU IIUis prescribed, and is calculated such that the stress ratio on the top surface has the prescribed value

22 12/

0 0

2 1 21 1for

A A

T dx T dx x B

V. Tvergaard, Int. J.Fracture (2009)

0 0

22 1 12 1 00 0

, for2 2A A

T dx T dx x BA A

periodicity conditions 1 1 2 2

1 2 1 2,u u u u

1 1 2 21 2 1 2,T T T T

/ 0.002 ,Y E 0.3 0.1N 0 0/ 4B A

average shear angle , defined by

tan /IU B


1 2 2 2 20( ) ( ) ,x x R 1 2 0 .T T The void surface, is initially stress free so that

If a hydrostatic pressure p was applied inside the voids to simulate the effect of crack surface contact in a relatively simple manner, the nominal tractions on the void surface would be

1 i ir ir ijk mrT p n g u g u , ,,2

r j j km k mT p n g u g u

: length of the void when it deforms into an ellipsoidal cross-section

:vV void volume per unit length in the direction3x

/ :vw V the average width of the void

Then, the average aspect ratio of the void is required to satisfy the inequality

/w In the expression due to hydrostatic pressure the traction component perpendicular to the line of length between the two

d i t f th id i i b d th t l th l th f th id i t i l d d h

1 2( i )T Tend points of the void is given by and the component along the length of the void is not included here. Then the nominal tractions applied to the void surface are

with . This load satisfies force equilibrium exactly. Small additional loading is applied to exactly satisfy moment equilibrium. Here there is no hydrostatic pressure, since the load components along the elongated void have been neglected to avoid that these components will tend to increase the length of the void.


1 2( sin cos ) T TiP

1 1 2( sin cos )sin P T T2 1 2( sin cos ) cos P T T

3 0P

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Fig. 2. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio , when , . and . Results for the transverse load (10)-(11) are compared with results for hydrostatic pressure loading inside the voids.

0 0/ 0.25R A 0.0

Fig. 3. Initial mesh and deformed meshes for , and . (a) Initial mesh. (b) At . (c) At . (d) At .

0 0/ 0.25R A 0.0 0.1

0.277 0.525 0.666

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Fig. 4. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio , when and .

0 0/ 0.25R A 0.6

Fig. 5. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio , when and .

0 0/ 0.20R A 0.0

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Fig. 7. Average shear stress vs. average shear angle for different values of the stress ratio , when and .

0 0

/ 0.25R A 0.20

Fig. 9. Deformed meshes for and . (a) For at . (b) For at . (c) For at .

0.0 0.15 0 0/ 0.25R A 0.529

0 0/ 0.20R A

0.611 0 0

/ 0.15R A 0.708

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Fig. 11. Average shear stress vs. average shear angle for different values of the initial yield strain , when , and .

/YE

0 0/ 0.25R A 0.15 0.0

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V. Tvergaard & K.L. Nielsen (2010)

Comparison with cell model study

Void volume fraction in a band of width equal to the void spacing is . 0f 02A

In the analysis for the damage model, the initial variation of the void volume fraction is taken to be

220 0 02f πR A

2

20

0

11 cos

2 2

xf x f

Afor otherwise

20 02 2 A x A 2 0f x

is the average shear angle for the cell analyzed, defined by

tan / IU B

a mechanism for ductile fracture in shear viggo...

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