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3.2 Some experiments concerning ductile failure
It is very important to understand conditions for specific mode of instability.
Manjoine from his experiments derived the following conclusions /11/:
– He defined ductility ࣧ and Davis triaxiality factor Tf as ࣧ ؝ ߝ ҧ ሺ
ߝ ҧሻ௨௡௜௔௫ , ܶ௙ ൌ ݎݐT ߪത (81)
and claims that they are roughly in inverse proportion; more precisely he assumedࣧ ൌ ݉݅݊൛1 ܶ௙ , ⁄ 2ଵି்೑ൟ ሺ82ሻ
i.e. lower of the two curves dependent on Tf .
– For elevated temperatures, long service life and small strain rates two major modes of
damage leading to failure hold: (a) crack initiation and growth by a maximum principal
stress history and (b) accumulation of shear strain up to strain instability.
– With reference to strength of grains and their boundaries mode (a) corresponds to grains
stronger than their boundaries (appearing at lower strain rates and higher temperatures),
whereas mode (b) is inverse to the above.
According to this, he suggested the following damage evolution equationࣧ ൌ ݂ ቄ෍ሺ ݐ௜ ݐ௥ ⁄ ሻ ఙ೘ೌೣ , ෍ሺߝ௜ ߝ כ ⁄ ሻఌሶ ,Tቅ (83)
where tr is rupture time at fixed maximum principal stress, ߪ௠௔௫ ߪ ؠெߝ ,כ is limit strain
at given strain rate, ߝሶ
, and true stress, T.
For a ductile alloy “D” ultimate strength (corresponding to maximal principal
engineering stress) decreases slowly with temperature for 0oF - 400oF; is practically
constant in the interval /400oF - 800 oF/ and then steeply decreases until 1400oF.
(Lindholm et al, /35/) performed biaxial tensile experiments (tension-torsion) on
beryllium and titanium specimens with following remarks:
– ductility is significantly reduced under biaxial tensile stresses, opposite to low-carbon
steels where critical subtangent Z = 2 for equibiaxial and Z = 1 for uniaxial tension;
– precompression could exhaust ductility and to lead to brittle fracture in subsequent
tensile loading (such a conclusion seems strange since compressive loading closes voids
and by plastic volume change should increase ductility);
– more generally strain history defined by diverse prestraining strain states (most
primitively by Bauschinger effect) influences limit strains very much;
– confirm mentioned importance of surface roughness on failure detection and grain size;
– ductility for the tested materials is maximal for pure shear, whereas smaller and equal
for uniaxial tension and equibiaxial tension but minimal for σ1/σ2 = 0.5; and
– they promote Mogi's criterion dependent on stress invariants as follows݂൫ߪത, ܶ௙ߪ ,௠௔௫, ߬௠௔௫൯ ൌ ሺܣ ൅ ܤሻߪതܶ௙ ൅ ߪܤതଶ ൅ ܥሺ߬௠௔௫ െ 2ߪ௠௔௫ሻ ൌ 1 (84)
where ߪ ௠௔௫ ߪ ؠଵ , ߬௠௔௫ ߪ ؠଵ െ ߪଷ , and ߪଵ ൐ ߪଶ ൐ ߪଷ are principal stresses.
Vavakin et al. /15/ investigated short term creep by staircase temporal stress loading
programs (Fig. 5. a and b) as well as by hold times at M–P and P–M nonproportional
paths in Hochenemser experiment (Fig. 5. c – vertices D 1, D2 as hold points). Their con-
clusion follows a previous experiments by Zhukov and Rabotnov /16/ and confirms
existence of short creep processes at abrupt change of stress trajectories.