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2. Considering a rectangular block whose sides are exposed to forces (Fig. 2.b) ܲሬሬሬሬԦఈ ൌ ܣ ఈݐԦ ఈ (Greek indices employed indicating no summation) induced by uniform
stresses ݐԦ௞ ൌ T· ݊ሬԦ௞ ؠ T·iԦ ௞ ൌ ߪ ௞௟iԦ௟
Hillier has derived the next expression for the critical subtangent1 ܼ ൌ 9 4 1ߪതଷ ቈ 1ܣఈ ߲ܲఈ௝ ߲݁௠௡ ߪ Ԣ௠௡ ߪԢ ௜௝ߜఈ௜ ൅ ߪఈ௝ߪԢ ௜௝ߪԢ ఈఉ ߜఈఉ ߜఈ௜቉ (17)
under the assumption that sides of the block are not influenced by shearing strains1
ܣఈ ߲ܣఈ ߲݁ఈఋ ൌ െ ߜఈఉ (18)
The application of critical subtangent formula lies practically in extremum values of
loads i.e. ݀ܲఈ ൌ 0 for diverse loading situations.
3. Let a rectangular block of dimensions ∆ܺଵ , ∆ܺଶ, ∆ܺଷ, (Fig. 2.b), be subject to
uniform plane stresses ߪଵଵߪ ,ଶଶ ൌ ݉ߪଵଵ ߪ ,ଷଷ ൌ 0, ߪଵଶ ൌ ݇ߪଵଵߪ ,ଶଷ ൌ 0, ߪଷଵ ൌ 0. Then
application of Eq. (17) to a Ramberg-Osgood material obeying Eq. (3), under critical
loads ߲ܲఈ௞ ߲݁௠௡ ⁄ gives the next value for critical subtangent /20/ܼ ൌ 4ሺ1 െ ݉ ൅ ݉ଶ ൅ 3݇ ଶሻଷ ଶ⁄
ሺ1 ൅ ݉ሻሺ4 െ 7݉ଶ ൅ 3݇ଶሻ (19)
which has been derived for case of normal strains i.e. for k = 0 by Swift (1957). In the
most special case of uniform tension, when m = 0 and k = 0 critical subtangent corres-
ponds to necking initiation (when engineering stress is maximum). Then Eq. (19)
specializes into Z = 1 .The corresponding “uniform" strain is easily derived in the sequel
(by means of ߪത ൌ ߪଵଵ ൐ 0) ݀ܲଵଵ ൌ ܣଵ݀ߪଵଵ ൅ ݀ ܣଵߪଵଵ ൌ ܲଵଵ ൬ ݀ߪଵଵ ߪଵଵ ൅ ݀ܣଵ ܣଵ ൰ ൌ 0
݀ ܣଵ ܣଵ ൎ െ݀݁ଵଵ ؠ െ݀ ߝ ҧ, ݀ߪത
ߪത ൌ ݊ߝҧ ฺ ൬ ݊ߝ ҧ
௨ െ 1൰ ݀ߝ ҧ ൌ 0
such that ߝ௨ ൌ ݊ (20)
Replacing this value into the definition of critical subtangent, Eq. (16) allows Z = 1. In
the special case when ݇ ՜ ∞ corresponding to pure shear ߪଵଵ ൌ ߪଶଶ ൌ 0; ߪଵଶ ് 0 ,
formula (19) fails, giving ܼ ՜ ∞ and the corresponding ߝ ҧ
௨ ՜ ∞. Suppose, following
Hill, Ref. /28/ that an initially anisotropic material deforms plastically according to the
following yield function2݂: ൌ 1 ݄ሺߝ ҧܲሻ T:ࣛ:Tൌ 1 ݄ሺߝ ҧܲሻ ࣛ௜௝௞௟ߪ ௜௝ߪ ௞௟ ؠ 2 3ߪതଶ
݄ሺߝ ҧܲሻ ൌ 1 (21)
where are: ࣛ -fourth rank tensor of constants depending on material symmetries (it will
be specified in the sequel for orthotropic as well as transversely isotropic materials),ߪത െequivalent stress, ߝ ҧ
௉ െ equivalent strain and h - scalar function describing “isotropic”
hardening, ߪത ൌ ߪതሺߝ ҧ
௉ሻ, capable to account only for homogeneous inflation of the yield
surface. Hill claims that ࣛ in general case is specified by six constants determined by
initial yield stresses for tensions and shears with respect to principal material directions.