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60
Tvergaard /24/ modified the above yield function by two coefficients, q1; q2 , as
follows ݂ ൌ ሺߪത ߪതெ ⁄ ሻଶ ൅ 2߱ݍଵܿ݋ݏ݄ ൬ 1 2 ࣲݍଶ ݎݐT
ߪതெ ൰ െ 1 െ ሺݍଵ߱ሻ ଶ ൌ 0 (70)
He suggested q1 = 1.5, q2 = 1 as an improvement.
A5-2. Void behaviour is governed by the following evolution߱ሶ ൌ ሺ1 െ ߱ሻߝݎݐ ሶ ൅ ࣜଵߝ ҧሶ
ெ ൅ ࣜଵߝ ҧ ሶ
ெ ൅ ࣜଶ܂ݎݐሶ (71)
where first term is responsible for void growth, while ࣜଵ ൐ 0, ࣜ ଶ ൐ 0 describe void
coalescence according to /24/. It is assumed here that void coalescence appears at some
critical ߱כ when instability abruptly leads to fracture. According to Gurson's assump-
tions ߱ כ ൌ 1, but Anderson, 1977, suggests ߱כ ൌ 0.25 for the growth of a spherical void
in the stress field in front of a crack tip. In the sequel ࣜଵand ࣜଶ are neglected (Gurson
assumed ࣜଵ ൌ 0, ࣜଶ ൌ 0).
A5-3. Equality of plastic work by matrix stress and strain versus work done by
average stress and strain leads toݎݐሼT ߝሶሽ ൌ ሺ1 െ ߱ሻߪതெ ߝ ҧ ሶ
ெ or ߪ௜௝ ߝሶ
௜௝ ൌ ሺ1 െ ߱ሻߪതெ ߝ ҧ ሶ
ெ (72)
A5-4. An associate flow rule is postulated in the common formߝ ሶ ൌר ߲݂ ߲T or ߝሶ
௜௝ ൌר ߲݂ ߲ߪ௜௝ (73)
In the standard way scalar function ר is found either from consistency condition ݂ ሶ ൌ 0
(rate independent plasticity) or it is a given function of stress and plastic strain
(viscoplasticity).
Analysis
In the sequel it is supposed that ࣲ ൌ 1, i.e. voids have approximately spherical shape.
Consider as in /18/ perfectly plastic matrix characterized by:ߪതெ ൌ ߪ଴ ൌ ܿ݋݊ݐݏ (74)
in plastic range. Then time differentiation of Eq. (69) combined with Eqs. (71) and (73)
gives us plastic modulus ר in the following form:רൌ ࣛଵ଴ߪሶ ൅ ࣛ ଶ଴܂ݎݐሶ (75)
where the coefficients are:ࣛଵ଴ൌ 2 ߪത
ߪതெଶ൤6 ߱ߪതெ ሺ1 െ ߱ሻሺ߱ െ ܥଵሻܵଵ൨ିଵ , ࣛଶ଴ൌ ߱ߪതெ ܵଵ ൤6 ߱ߪതெ ሺ1 െ ߱ሻሺ߱ െ ܥଵሻܵଵ൨ିଵ (76)
and denotations used ܵଵ ݏ ؠ݄݅݊ ൬ ݎݐT 2
ߪതெ ൰ , ܥଵ ݏ ؠ݄݅݊ ൬ ݎݐT 2
ߪതெ ൰ . (77)
The governing differential equations are obtained inserting (75) in Eqs.(71) and (73)ߝ ሶ
ଵ ൌ ൫ࣛଵ଴ߪത ൅ሶ ࣛ ଶ଴ݎݐTሶ ൯ ቆ 2ߪଵ െ ߪଶ ߪതெଶ ൅ ߱ܵଵ ߪതெ ቇ , (78)
ߝሶ
ଶ ൌ ൫ࣛଵ଴ߪത ൅ሶ ࣛ ଶ଴ݎݐTሶ ൯ ቆ 2ߪଶ െ ߪଵ ߪതெଶ ൅ ߱ܵଵ ߪതெ ቇ , (79)