IFMASS10

across (leading to normal fracture) or along a typical ear shape hole (leading to

delamination). The third mode of shear type is not considered in his paper /23/.

Assumptions

A1-1: Rate independent rigid plastic material is assumed which obeys associativity of

flow rule based on Mises yield surface. In this case the following equation holds in terms

of true stress deviator and total true strain deviator

ᇱ

ൌ ݀



ᇱ

, ݀



ൌ 3 2 ݀

തതത

ത

(1)

where “rigidity” means



(i.e. total and plastic strains are assumed to be

approximately equal for advanced plastic strain at fracturing process), and

ଶ

3 2ൗ

ᇱଶ

ଶ

ൌ 2 3ൗ

ݎݐ

ᇱଶ

Are equivalent stress and equivalent increment of total strain (increment of plastic

accumulation path), respectively.

A1-2.

Three mutually perpendicular elliptically cylindrical holes (Fig. 1.a) are assu-

med to exist such that each hole surrounded by a solid material with cross section of

diameters equal to average spacing between holes forms a cell.

a. Three sets of mutually perpendicular elliptical cylindrical b. McClintock's representative

holes annular cylinder

Figure 1: An illustrative damaged specimen

For instance, for a cell whose axis is with growth extending along axis

, the

corresponding relative growth factor is defined as follows

௭௕

: ൌ ൬ ܾ ݈

௕

൰ / ൬ ܾ

଴

௕଴

൰

ܨ ؠ

௭௕

௭௕଴

ൗ

௭௕ ௙

ൌ 1 2⁄

, then the

zb-

fracture begins due to coalescence of

-holes into

direction

and

௭௕௙

ൌ

௭௕ ௙

௭௕଴

ൗ ൌ 1 2⁄

௭௕଴

⁄

. In general, for the whole set

௜௝

ൌ

௜௝

௜௝଴

⁄

the

fracture begins when

݉ܽ

௜,௝

൫

௜௝

൯ ൌ 1 2⁄

Finally McClintock proposed a logarithmic, additive and normalized, damage measure





௜௝

ൌ ݈݊ ׎

௜௝

݈݊׎

௜௝ ௙

ൗ

(2)

which at fracture became unity. Its additivity is guaranteed only if Lagrangian approach

with fixed reference configuration is accepted. Otherwise in the terminology of FEM, if

updated (i.e. variable) reference configuration is employed, then it is not additive any

more. A modified damage measure