IFMASS10

ு

ൌ 1 √3 √2 ൅ ܾ ൅ ܿሺܿ െ 2݉ ൅ ܾ݉

ଶ

ሻ

ଷ ଶ⁄

ଶ

൅ ܾ

ଶ

ଷ

െ ݉ሺ2ܿ െ 1ሻ െ ݉

ଶ

ሺ2ܾ െ 1ሻ

differing from the corresponding Korhonen's expression by

√2 ൅ ܾ ൅ ܿ √2ܾ ൅ 2ܿ െ 2

⁄

, a

constant, which in the case of isotropy becomes 1. Accordingly, uniform (limit) strain

concordant with the above expressions differs from Korhonen's limit strain also up to this

constant. In the case of isotropy

b = c =

2 both limit strains coincide.

Here one may draw following conclusions:

–

the method is easy for application and gives good results for bodies whose characte-

ristic dimensions are all of the same order of magnitude;

–

its inability to account for strain rate effects is a serious shortcoming;

–

the assumption about simultaneous maxima of engineering in-plane principal stresses

should be checked by experiments (some experimental data, in fact, demonstrate that

indeed their maxima are not simultaneous -Albertini /6/).

2.3. Localized necking in thin sheets

Physically, the formation of a neck in a sheet requires deformation in a thin zone

where the material does not change in length along the zone because of the constraint of

rigid shoulders on each side, Ref. /2/.

–

The main contribution to be considered here is that of Hill /29/.

–

Its application to anisotropic sheets was done in /21/.

Assumptions

A3-1. Let a necking strip of width B appears in a sheet with velocity of its propagation

inclined by an angle

(Fig. 3), Hill /29/. Assumption rigid plasticity (i.e.negligible

elastic strain) and plastic incompressibility (cf. A2-3) deformation gradient reads:

Fൌ ൝ 1 ൅

ݐݏ

0 0 1 ሺ1 ൅

ݐݏ

ሻ

⁄

ൡ

(35)

Here, material coordinate frame is chosen to coincide

-axis with the strip, whereas

ؠ ݏ

݅݊߰

⁄ , ܿ

߭ ܿ݋

.ܤ

⁄

Then eigenvalues of stretching tensor

Dൌ

ሶ

are

ሶ

ଵ

ൌ ߭ሺ1 ൅

ሻ

ሺ1 ൅

ݐݏ

ሻ ,

ሶ

ଶ

ൌ െ ߭ሺ1 െ

ሻ

ሺ1 ൅

ݐݏ

ሻ ,

ሶ

ଷ

ൌ ߭

ܤ ݏ

ሺ1 ൅

ݐݏ

ሻ

(36)

Hill in /29/ neglects the bracket in the denominators under the assumption that

ݐݏ

.The proper directions of

ሶ

are oblique with respect to the strip by angle of

ሺ2

െ ߰ሻ 4⁄

. Orientation of velocity discontinuity found from the above relationships:

݅݊߰ ൌ

ሶ

ଵ

൅

ሶ

ଶ

ሶ

ଵ

െ

ሶ

ଶ

(37)

delimits applications of this method only to time instants when:

ሶ

ଵ

൐ 0,

ሶ

ଶ

൏ 0.

(38)

A3-2. For localized necking in sheets

ห

ሶ

ଵ

ሶ

ଷ

⁄ ห

and

ห

ሶ

ଶ

ሶ

ଷ

⁄ ห

such that

maximum of largest principal engineering stress gives

ሶ

ଵ

ൎ െ

ଶ

ሶ

ଷ

ଶ

ଷ

ൌ െ

ሶ

ଷ

(39)