A MECHANISM FOR DUCTILE FRACTURE IN SHEAR. Viggo Tvergaard Department of Mechanical Engineering,

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1 A MECHANISM FOR DUCTILE FRACTURE IN SHEAR Viggo Tvergaard Department of Mechanical Engineering, Solid Mechanics Technical University of Denmark, Bldg. 44, DK-8 Kgs. Lyngby, Denmark N.A. Fleck, J.W. Hutchinson & V. Tvergaard, JMPS (1989) 1

2 N.A. Fleck, J.W. Hutchinson & V. Tvergaard, JMPS (1989) V. Tvergaard, Int. J. Mech. Sci. (8) Fig. 1. Periodic array of cylindrical voids used to model ductile failure in shear.

3 conditions of simple shear J flow theory ij ij G/ g ij 1/Gij g ij ijk ij L k / E, Y / / 1/ N Y E Y, Y Mises stress ij (3 ss / ) 1/ e ij 1 k ij ui, j uj, i u, iuk, j ij ij k i ij i ij, i V k, j A i V ij A i u u dv T u da dv T u da u U, u U for x B 1 I II u U, u U for x B 1 I II U I is prescribed, and U II is calculated such that the stress ratio on the top surface has the prescribed value / 1 A 1 Tdx1, 1 1 Tdx1 for x B A 1 A A A A periodicity conditions u 1 1 u, u 1 u T 1 1 T, T 1 T Y / E.,.3 N.1 B / A 4 average shear angle, defined by tan UI / B 3

4 The void surface, ( x ) ( x ) R, 1 is initially stress free so that T T 1. At a later stage of the deformation a hydrostatic pressure p is applied inside the voids to simulate the effect of crack surface contact in a relatively simple manner. The nominal tractions and their increments on the void surface are T i p ir n, T i p ir p ir n r ir 1 ijk mr g u g u j j km k, m, r : length of the void when it deforms into an ellipsoidal cross-section Vv : 3 void volume per unit length in the x direction w V 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. 4

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7 V. Tvergaard, Int. J.Fracture (9) u U, u U for x B 1 I II U I is prescribed, and U II u U, u U for x B 1 I II is calculated such that the stress ratio on the top surface has the prescribed value / 1 A 1 Tdx1, 1 1 Tdx1 for x B A 1 A A A A periodicity conditions u 1 1 u, u 1 u T 1 1 T, T 1 T Y / E.,.3 N.1 B / A 4 average shear angle, defined by tan UI / B The void surface, ( x ) ( x ) R, 1 is initially stress free so that T T 1. 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 i ir T p n, ir 1 ijk mr g u g u,, r j j km k m : length of the void when it deforms into an ellipsoidal cross-section Vv : 3 void volume per unit length in the x direction w V 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 1 end points of the void is given by ( T sin T cos ) and the component along the length of the void is not included ddhere. i Then the nominal tractions P applied to the void surface are 1 1 P ( T sin T cos )sin 1 P ( T sin T cos ) cos 3 with P. 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. 7

8 Fig.. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio, when, R / A..5 and.. Results for the transverse load (1)-(11) are compared with results for hydrostatic pressure loading inside the voids. Fig. 3. Initial mesh and deformed meshes for R / A.5,. and.1. (a) Initial mesh. (b) At.77. (c) At.55. (d) At

9 Fig. 4. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio, when R / A.5 and.6. Fig. 5. Average shear stress vs. average shear angle for different values of the limiting void aspect ratio, when R / A. and.. 9

10 Fig. 7. Average shear stress vs. average shear angle for different values of the stress ratio, when R / A.5 and.. Fig. 9. Deformed meshes for. and.15. (a) For R / A.5 at.59. (b) For R / A. at.611. (c) For R / A.15 at.78. 1

11 Fig. 11. Average shear stress vs. average shear angle for different values of the initial yield strain Y / E, when R / A.5,.15 and.. 11

12 V. Tvergaard & K.L. Nielsen (1) Comparison with cell model study Void volume fraction in a band of width equal to the void spacing A is f πr A. f In the analysis for the damage model, the initial variation of the void volume fraction is taken to be f x f 1 x 1 cos A for A x A otherwise f x is the average shear angle for the cell analyzed, defined by tan U / B I 1

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