IFMASS10

52 / 352

48
ߟ୸ୠ ln ׎୸ୠ୤ ൌ ݀ ln ൬ 1 െ ݁ 1 െ ݁଴൰ െ ݀ ߝ௕ (10)
Neglecting influence of eccentricity, i.e. the second term on right-hand-side (RHS) for
constant stress state we have݀ߟ௭௕ ݀ߝ ҧ ln ׎ ௭௕௙ൌ √3 2ሺ1 െ ݊ሻ ݏ݄݅݊ ቈ √3ሺ1 െ ݊ሻ 2 ߪ௔ ൅ ߪ௕ ߪത ቉ ൅ 3 4 ߪ௔ െ ߪ௕ ߪത (11)
Let either stress components are constant or a proportional stress path is considered.
Then the above equation is integrated up to value ௭௕ ௙ൌ 1 to give equivalent fracture
strain when z-holes coalesce in b-direction ߝҧ௙ ൌ ln൫׎௭௕௙൯ ቆ √3 2ሺ1 െ ݊ሻ ݏ݄݅݊ ቈ √3ሺ1 െ ݊ሻ 2 ߪ௔ ൅ ߪ௕ ߝҧ ቉ ൅ 3 4 ߪ௔ െ ߪ௕ ߪത ቇ (12)
Assuming in the case of plane tension െ1 ൏ ݉ ߪ ؠ௕ ߪ௔ ൏ 1 ⁄, corresponding to
triaxiality factor, /11/ 0 ൏ ܶ௙ ൌ tr T ߪത⁄ ൏ 2 (13)
the above equation permits graphical representation of fracture strain derived by critical
growth factor versus triaxiality. It should be noted that triaxiality factor for the case of
isotropy and plane stress by its values delimits some characteristic regions:ܶ௙ ൌ 0, ሺߪଵ ൌ െߪଶሻ -simple shear, ܶ௙ ൌ 1, ሺߪଶ ൌ 0ሻ -uniaxial tension, ܶ௙ ൌ √3, ሺߪଵ ൌ 2 ߪଶ ሻ -plane strain, ܶ௙ ൌ 2, ሺߪଵ ൌ ߪଶሻ -equibiaxial tension.
According to McClintock's results /4/ and his criterion the smallest fracture strain is
obtained for equibiaxial stress state and the largest for shear when ߪ௕ ൌ െߪ௔.
2.2 Diffuse plastic instability
According to Ref. /2/ a nonuniform strain field may develop twofold: (a) thinning
during tension loads occurs very gradually in dimensions comparable with specimen
dimensions and, (b) it occurs in a region comparable with sheet (or specimen) thickness.
The first is called diffuse instability whereas for the second phenomenon the name
localized instability is chosen. Due to above distinction the diffuse instability could
appear mainly when cylindrical specimen are used.
Assumptions
A2-1.The material considered is assumed to be strain hardening rigid plastic obeying
assumptions A1-1 and A1-3. Moreover, multiaxial stress state is restricted to be
homogeneous, i.e. uniform in terminology of Ref. /20/.
A2-2. The basic assumption lies in the principle of maximum rate of plastic work as
stated in Ref. /19/: ׬ ሺT'௔௖௧௨௔௟ െ T'ሻ: ߝሶ
௔௖௧௨௔௟ᇱ ܸ݀ ൒ 0
௏ (14)
According to Hill's terminology throughout this subsection the subscript “actual” is
used to denote actual or true quantities like stress and strain (the word “actual” is used
here instead of “true” to avoid confusion). Likewise T௔௖௧௨௔௟ᇱ and ߝ௔௖௧௨௔௟ᇱ are actual stress
and actual strain deviators, respectively. These two must be related through the strain