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Influence of defect shape on damage evolution and fracture behavior of ductile materials
Engineering Fracture Mechanics Vol. 51, No. 3, pp. 337-347, 1995 Copyright © 1995 Elsevier Science Ltd
Pergamon
0013-7944(94)00254-1
INFLUENCE AND
OF DEFECT
FRACTURE
SHAPE
BEHAVIOR
Printed in GreatBritain.All rightsreserved 0013-7944/95 $9.50+ 0.00
ON DAMAGE OF DUCTILE
EVOLUTION MATERIALS
SUN YI Department of Space Technology and Mechanics, Harbin Institute of Technology, Harbin, 150001, P.R. China C. ERIPRET and G. ROUSSELIER Electricite de France, Departement MTC, Les Renardieres, 77250 Moret sur Loing, France
Abstract--A generalized standard material model is developed for ductile materials containing aligned defects with arbitrary shape under self-similarexpansion based on the work of Nguyen and Bui [Sur les materiaux elastoplastiques a ecrouissage positif et negatif. J. Mecanique 13, 321 (1974)] and Rousselier [Finite deformation constitutive relations including ductile fracture damage, in Three-dimensional Constitutive Equations and Ductile Fracture (Edited by S. Nemat-Nasser), pp. 331-355, North-Holland (1981)]. The influence of defect shape on damage evolution and instability of the material is investigated. The finite elementcalculation is made for notched and cracked specimens,with mierovoid and microcrack damage. The result shows that the defect shape has significantinfluence on the fracture initiation and crack growth behavior. The present model is also applied to predict the J-resistance curve of a CT specimen for an aged duplex stainless steel, and the results are in good agreement with the experiment.
1. I N T R O D U C T I O N THE FORMATION, growth and coalescence of micro defects is the chief damage and fracture mechanism of engineering materials. For ductile materials, such a kind of damage is most represented by microvoids and microcracks. Several models were proposed to describe the mechanical behavior of voided materials [1-5]. For microcrack damage, He and Hutchinson [6] gave a perturbation solution of an isolated penny-shaped crack in an infinite body of power-law materials. Guennouni [7] made the finite element calculation for a finite cell containing a plane crack. These works were based on microscopic analysis of the damage process. Another approach, which is referred to as the continuous damage theory, is based on the continuum thermodynamics theory in which the material damage is described by an internal variable. A m o n g these the generalized standard material (GSM) theory developed by Nguyen and Bui [8] and Rousselier [9] is of particular interest. A standard methodology [10] for analyzing the microvoid damage and fracture process was proposed. However, less work has been done for ductile material with microcrack damage. Here, we first generate a G S M formulation for ductile material containing aligned defects with arbitrary shape under self-similar expansion. Then, we investigate the influence of defect geometry on the macromechanical behavior of the material. Next, we make finite element calculations to examine the damage and fracture process in notched and cracked specimens with different defect shapes, and compare with the experiment.
2. G S M F O R M U L A T I O N Consider a macroscopic material element containing distributed defects (Fig. 1). Let v be the velocity field within the element, V and S the volume and the outer surface of the element, Sk the surface of the kth defect and n the unit outward normal vector of the integral surface. The macroscopic deformation rate Dq is written as
Do = V1
½(v,nj+vjni)ds
1
=--.~
Vn
dodV +
½(v,nj+vjn,)ds
k = l JS k
=a*+~
E f~,
k=l
(1)
where d is the deformation rate in the matrix, d* its volume average and ~,~ the damage evolution tensor. For the self-similar expansion 337
338
SUN YI et al. S
V Fig. 1. An element of material containingdistributed defects. k k VaoJo.,
(2)
where V~ is the volume of the kth defect, co~ is time-invariant and describes the influence of the shape and orientation of the defect. In the case of aligned parallel defects D o = d* +
- ~ ~%,
(3)
where M
l?u = E I?~.
(4)
k=l
For an incompressible matrix Ogkk= 1.
(5)
From eq. (3) we see that for aligned parallel defects under self-similar expansion, we can use a scalar variable to describe the damage process. We consider the matrix as a homogeneous isotropic hardening material. Its hardening behavior is described by an internal variable ~. The damage effect [eqs (1)-(3)] could be represented by a scalar internal variable ft. The specific free energy is supposed to have the following form (p(E~, 0~,/~)
=
¢p, (e,~) + ¢P2(~, fl),
(6)
where E,~ is the elastic strain, 1 e e ~o~(E eu) = ~C~ktE eE ki
(7)
is the elastic recoverable energy and CUkt the elastic moduli. The plastic potential is F ( Z o, A, B)
= ')"~'eq"q- A "3I- B e ( ~'~*, A ),
(8)
where
1
AB=
0qh O~P2
E* = £um U =
[3~-~/'$--,t ~1/2
Z~ = IF,o - ½Zkk 6 o
(9)
Influence of defect shape on ductile materials
339
0.2
0.1
0
0.2
0.4
0.6
0.8
I
1.0
Fig. 2. Instability strain e~ vs stress-state paramater 7. - - Microcrack damage model.---Microvoid damage model. p = p(fl) is the density, a the Cauchy stress, and Z, A and B are generalized forces corresponding
to e e, ~ and ft. Notice that in eq. (6) we do not assume the separability of ~ and fl, so that A and B are general functions of ~ and ft. This stems from the fact that in many cases g depends also on the hardening behavior of the material, and ~o2(~,/~) = f l (~) +fz(]~)
(10)
will be considered as a special case. With the use o f the first and second t h e r m o d y n a m i c s laws, and the a s s u m p t i o n of n o r m a l dissipativity [9] we get Z~ = C~ktee~ 8F
(3Z~
OF
(
t?g
)
Og)
I+BSj
8F
(11)
51-_~
The law of mass conservation yields
IJ + pDPkk = O,
(12)
31 I
21
1 0
I 0.2
I 0.4
I 0.5.
I 0.8
I
1.0
Fig. 3. Maximum stress E m vs stress-state parameter 7. - - Microcrack damage model. - - - Microvoid damage model.
SUN YI et al.
340 1.6-
1.2- ~'~
,..g
0.8 -
",..,
0.4~
0
~.,..
I 0.2
J 0.4
1 0.5
f 0,8
Fig. 4. Instability value of damage /~c vs stress-state parameter ~/. - - - - Microvoid damage model.
I 1.0 Microcrack damage model.
f r o m eq. (11) tgg
(13)
0 It follows that 1
dp
Bp dfl
1 ~g
1
g ~Y~*
(14)
a* '
w h e r e a* = a*(A, B). S o l v i n g eq. (14) g = go e x p ~-~
B=
a* dp
-
(15)
a'q,
p dlJ
w h e r e go is t a k e n a s a c o n s t a n t , a n d 1 dp q = --___
p d#
(16)
= q(fl).
1000~
75C
b
250
0
I 0.05
I 0,10
I 0.15
I 0.20
Fig. 5. Uniaxial stress-strain curve of an aged duplex stainless steel.
Influence of defect shape on ductile materials
2~/R--10 -
-
R=2
-
~
ii~ 21-I ?
90
4
341
AE2 _ I
(a)
25.0 -
(b) 22.5 20.0
\/VVV
17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0 -2.5 -2.5
I
I
I
I
0
2.5
5.0
7.5
I
I
I
[
I
I
1 0 . 0 12.5 15.0 17.5 20.0 22.5 25.0
Fig. 6. The AE2 notched axisymmetric specimen. (a) Specimen geometry, (b) finite element mesh.
Let AI=A,
Az=B,
el=a,
a2=fl.
(17)
The consistency condition becomes OF Z'J+b-Z 0F Ao=0,
(18)
where i,j = 1-3 and u = 1, 2. From eqs (8), (9) and (11) ~A.e ~
. a2¢p2 OF
(19)
so that = _ 1_ _O_ F . H OZq E~'
(20)
OF OZqh OF H =-OA. c7% Oe,, OA~, "
(21)
where
The rate-type constitutive equation can be written as Y-,~= Lij~lDkl,
(22)
342
SUN YI et al.
where OF OF Cijmn ~ m n Cklpq ~ p q
(23)
Lijkl = Cijkl --
H + a~nmnCmnpq We will discuss some special forms o f ~02. (1) a * = a0 = constant. In this case ao dp B . . . . .
p
thus
d~
f*(fl)
=
Oq~2 ---,
(24)
O8
["
8) =
--J f * ( f l ) dfl + f l (~) = f l (ct) + f : ( f l ) .
(25)
(2) a* = ao - kA. F r o m
e=
~-
(26)
we get the following partial differential equation (27)
P ~-~ + O ~-~ = R,
where P = kq,
Q = I,
(28)
R = -aoq,
The general solution of eq. (28) is
(29)
f ( u , v) = 0 ,
where u, v are Lagrange auxiliary functions u= ~ -
dfl = c~ + k ln p + c~
v = ~02 -
(30)
dfl = q~2 - ~0 In p + c2.
3. CONSTITUTIVE BEHAVIOR (1) D a m a g e variable Theoretically, the choice o f damage variable can be an arbitrary function of p. However, some results o f micromechanical analysis may serve as a guide. F r o m eq. (11) Og Z*' DPq = 211 + 2B 0E* ]~eq
~-2(c02- ½)(B 0g "~21'/2
where
(31)
(32)
~*' = ]E~e;~j, 092 = og~e~O. F o r dilute distribution of defects, it is reasonable to assume that
(33)
B 0~, < 1 , thus
ag
D~k = 2B ~
= DPq
go exp
.
(34)
Influence of defect shape on ductile materials
16.o (a)
16.0
12.0
12.0
8.0
8.0
4.0
343
(a)
4.0
"'"-0.2 "0.4 0.6
1 . 0 ~ - -
0.0
0.0
I 0.0
I 4.0
I 8.0
I 12.0
I 16.0
I
0.0
I
4.0
I 8.0
I 12.0
I 16.0
16.o 2h) 16.0
(b) 12.0
12.0 8.0 g.O 4.0
~1.0
4.0 ""0.9
~3"05.0 7 ~" 0.0
•1 1 .
0.0
[
I 0.0
I
4.0
I
8.0
I
12.0
0.0
I
I
4.0
I
8.0
I
12.0
I
16.0
16.0 Fig. 8. Microcrack damage evolution in AE2 notched specimen. (a) At 30 step loading, (b) at 40 step loading.
Fig. 7. Microvoid damage evolution in AE2 notched specimen. (a) At 30 step loading, (b) at 40 step loading.
Comparison with the microscopic analysis prediction [3, 6] gives D~k = DPq(1 - p)f(Y~*, A ),
(35)
it is suggested that 1
dp
pdfl
1 -p
(36)
(1 - p)lgl
~ 1,
(37)
and eq. (33) becomes so that 1
P = 1 +exp(fl --fl0)
=-po)/~,
flo : l n ( ~ p°
(38)
344
S U N Y I et al. P
A
C) _L
0
ao
-r
bo
A'
W P Fig. 9. The c o m p a c t tension (CT) specimen. W = 50 m m , a 0 = 30 m m , b o = 20 ram.
where Po is the initial density. (2) The hardening law We choose here the power-hardening law A -
-a~ 1+~
(39)
,
where
+ ln( t.
(40)
Employing eq. (39) in eq. (21) gives
. = NOy l u)N(l + k q g u0 k + ~
~-
1
-a
.2dq g -~.
(41)
(3) Material instability We consider an element of material subject to monotonous loading. The matrix material is supposed to be plastically incompressible. The criterion of material instability caused by damage development is given by Drucker's inequality ~-,~jDC:= 22H ~
(42)
The calculation is carried out under axisymmetric loading, with different stress triaxiality E22 = E33 = l / E l l ,
E 0 = 0,
i ¢=j.
(43)
To exhibit the influence of defect shape on damage evolution and macro behavior of materials, we deal with two typical damage mechanisms, microvoid damage and microcrack damage. For microvoid damage
% = ½60,
(44)
090 = ninj,
(45)
while for microcrack damage where n is the unit normal vector perpendicular to the crack surface. In the present calculation, we assume that n coincides with the maximum principal stress direction. The material parameters chosen in the calculation are
Influence of defect shape on ductile materials
345
70 60
50
40
30
Ilii11111
I
,L_L/A
! I Ill I I I I l l l [ | l l I Jllllllill!lllll
[ I I ~
20-
10-
I I I
0-10 -30
I -20
I -10
[ 0
I 10
I 20
L 30
[ 40
I 50
Fig. 10. The finite element mesh for CT specimen.
ay = E
u0 = 0.003,
v = 0.3,
N --- 0.25.
(46)
The damage parameters for the two damage mechanisms are chosen to produce the same E ~ curve in uniaxial tension. For microvoid damage 1 - P0 = 0.03,
a* = 0.64ay
(47)
a* = 2ay.
(48)
and for microcrack damage 1 -
Po =
0.01,
Figure 2 shows the variation of the instability strain along the X~ axis with the stress triaxiality. It is interesting to notice that the difference of the variations of instability strains of the two damage models may reach as much as 100%. What is more surprising is the change of the maximum stress, in which the two models predict qualitatively different results (Fig. 3). This indicates that for notched and cracked specimens, the damage will be localized more seriously in the microcracking model than in the void growth model. Further discussion on this point will be given later. On the other hand, the critical value of the damage variable fic predicted by the two models does not show much difference (Fig. 4). It should be noted that such variation of the stress state may also be found in the same damage mechanism with a different damage hardening coupling effect [eq. (30)]. These results signify that the accurate damage evolution description is not only necessary for local damage prediction, but also for the macroscopic material behavior characterization. 4. FRACTURE P R O C E S S ANALYSIS We apply the present theory to study the fracture process in notched and cracked specimens. The realization and calculation of the present model are carried out on EDFs program ALI-BABA, with the Lagrange large deformation formulation. The material parameters used in the calculation are based on an aged duplex stainless steel [11]. The experimental result of the uniaxial stress-strain relation is shown in Fig. 5. The finite element calculation is focused on the typical damage mechanisms. The initial values of the damage parameters are determined from an AE2 axisymmetric specimen, according to the standard procedure described in ref. [10]. For the void growth mechanism 1 - P0 = 10-3,
ty* = 220 MPa.
(49)
346
S U N YI et al.
Node A
cJ
tm
Failure p "
O
Loading Fig. I 1. The failure of the element in front of the main crack.
For the microcracking process 1 - P0 = 10-6,
o'* = 250 MPa.
(50)
The geometry of the specimen and the mesh used in the calculation are shown in Fig. 6. Figures 7 and 8 show the damage distribution in the notched specimen. It is shown that for the void growth mechanism, the damage begins at the root of the notch, then with the increase of the loading the maximum damage region moves towards the specimen center. Fracture begins in the central region of the specimen. This coincides with the "cup and core" fracture surface observed in experiment. However, for the microcracking process the damage always concentrates at the notch root and the fracture first takes place near the root surface. Such a phenomenon was observed in the notched specimen test of aged duplex stainless steel. Based on the above analysis, we investigate further the influence of defect shape on crack growth behavior in a CT specimen (Fig. 9). The eight-node quadrilateral isoparametric elements are employed near the crack tip and over most parts of the specimen. The mesh used in the calculation is shown in Fig. 10. It should be mentioned that one of the special features of the present model is that the fracture initiation and the crack growth can be simulated without artificial interference. With the increase of loading, material instability takes place at the element in front of the main crack which is mostly damaged, the damage localizes and the element stiffness drops toward zero, following eq. (40), thus forming the crack growth. Here, we define the failure of the element as the sudden increase of the opening displacement of the node just above the crack tip (Fig. 11). Since the damage localization depends on the mesh size, we choose the element length
200
10G
0
I
I
I
I
2
3
Aa (ram) Fig. 12. Comparison of the predicted J - R curves for CT specimen with the experiment. damage model. - - - Microvoid damage model. • Experiment.
Microcrack
Influence of defect shape on ductile materials
347
in front of the crack such as to get the prescribed JI, value. In the present calculation this length is taken as 0.55 mm, which agrees with the experimental result of the aged duplex stainless steel. The predicted J-resistance curves for the two damage mechanisms are shown in Fig. 12. The difference is significant. Such different crack growth behavior might be explained by the difference of the two damage mechanisms in the instability and the damage localization behavior, as previously mentioned. These results emphasize again the importance of the defect shape on damage evolution and macro fracture properties of the material. A comparison is also made between the predicted J - R curves and the experimental result for the aged duplex stainless steel (Fig. 12). It is found that the microcrack damage model could give quite a satisfactory result, which also confirms the prediction of the location of fracture initiation in notched specimens. 5. CONCLUSIONS The characterization of the micro defects and damage development is very important in obtaining the accurate macroscopic material response. The present study reveals that the defect shape has significant influence on damage evolution and fracture behavior of the material. This might be important in engineering design. It should be supplemented that the present model is an idealized one and certain simplifications are made in the calculation. Some factors, e.g. defect rotation and non-similar evolution are not considered. These are left for further research. REFERENCES [1] F. A. McClintock, A criterion for ductile fracture by the growth of holes. J. appl. Mech. 35, 363 371 (1968). [2] J. R. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201-217 (1969). [3] B. Budiansky, J. W. Hutchinson and S. Slutsky, Void growth and collapse in viscous solids, in Mechanics o['Solids (Edited by H. G. Hopkin and M. J. Sewell), pp. 13 45. Pergamon Press (1982). [4] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. Part l--yield criteria and flow rule for porous ductile media. J. Engng Mater. Technol. 99, 2-15 (1977). [5] Sun Yi and Wang Duo, A lower bound approach to the yield loci of porous materials. Acta Mech. Sinica 5, 237 243 (1989). [6] M. Y. He and J. W. Hutchinson, The penny-shaped crack and the plane strain crack in an infinite body of power-law materials. J. appl. Mech. 48, 830-840 (1981). [7] T. Guennouni, Plastic and viscoplastic behavior of cracked materials. Eur. J. Mech. 8A, 491-511 (1989). [8] Q. S. Nguyen and H. D. Bui, Sur les materiaux elastoplastiques a ecrouissage positif et negatif. J. Mecanique 13, 321 (1974). [9] G. Rousselier, Finite deformation constitutive relations including ductile fracture damage, in Three-dimensional Constitutive Equations and Ductile Fracture (Edited by S. Nemat-Nasser), pp. 331 355. North-Holland (1981). [10] G. Rousselier, Ductile fracture models and their potentials in local approach of fracture. Nul. Engng Des, 105, 97 111 (1987). [11] S. Bonnet, et al., Evolution of mechanical properties of various cast duplex stainless steels in relation to metallurgical and aging parameters: an outline of current EDF programs. International Workshop on Intermediate Temperature Embrittlement Process in Duplex Stainless Steels, Oxford (1 2 August 1989). (Received 18 March 1994)
Pergamon
0013-7944(94)00254-1
INFLUENCE AND
OF DEFECT
FRACTURE
SHAPE
BEHAVIOR
Printed in GreatBritain.All rightsreserved 0013-7944/95 $9.50+ 0.00
ON DAMAGE OF DUCTILE
EVOLUTION MATERIALS
SUN YI Department of Space Technology and Mechanics, Harbin Institute of Technology, Harbin, 150001, P.R. China C. ERIPRET and G. ROUSSELIER Electricite de France, Departement MTC, Les Renardieres, 77250 Moret sur Loing, France
Abstract--A generalized standard material model is developed for ductile materials containing aligned defects with arbitrary shape under self-similarexpansion based on the work of Nguyen and Bui [Sur les materiaux elastoplastiques a ecrouissage positif et negatif. J. Mecanique 13, 321 (1974)] and Rousselier [Finite deformation constitutive relations including ductile fracture damage, in Three-dimensional Constitutive Equations and Ductile Fracture (Edited by S. Nemat-Nasser), pp. 331-355, North-Holland (1981)]. The influence of defect shape on damage evolution and instability of the material is investigated. The finite elementcalculation is made for notched and cracked specimens,with mierovoid and microcrack damage. The result shows that the defect shape has significantinfluence on the fracture initiation and crack growth behavior. The present model is also applied to predict the J-resistance curve of a CT specimen for an aged duplex stainless steel, and the results are in good agreement with the experiment.
1. I N T R O D U C T I O N THE FORMATION, growth and coalescence of micro defects is the chief damage and fracture mechanism of engineering materials. For ductile materials, such a kind of damage is most represented by microvoids and microcracks. Several models were proposed to describe the mechanical behavior of voided materials [1-5]. For microcrack damage, He and Hutchinson [6] gave a perturbation solution of an isolated penny-shaped crack in an infinite body of power-law materials. Guennouni [7] made the finite element calculation for a finite cell containing a plane crack. These works were based on microscopic analysis of the damage process. Another approach, which is referred to as the continuous damage theory, is based on the continuum thermodynamics theory in which the material damage is described by an internal variable. A m o n g these the generalized standard material (GSM) theory developed by Nguyen and Bui [8] and Rousselier [9] is of particular interest. A standard methodology [10] for analyzing the microvoid damage and fracture process was proposed. However, less work has been done for ductile material with microcrack damage. Here, we first generate a G S M formulation for ductile material containing aligned defects with arbitrary shape under self-similar expansion. Then, we investigate the influence of defect geometry on the macromechanical behavior of the material. Next, we make finite element calculations to examine the damage and fracture process in notched and cracked specimens with different defect shapes, and compare with the experiment.
2. G S M F O R M U L A T I O N Consider a macroscopic material element containing distributed defects (Fig. 1). Let v be the velocity field within the element, V and S the volume and the outer surface of the element, Sk the surface of the kth defect and n the unit outward normal vector of the integral surface. The macroscopic deformation rate Dq is written as
Do = V1
½(v,nj+vjni)ds
1
=--.~
Vn
dodV +
½(v,nj+vjn,)ds
k = l JS k
=a*+~
E f~,
k=l
(1)
where d is the deformation rate in the matrix, d* its volume average and ~,~ the damage evolution tensor. For the self-similar expansion 337
338
SUN YI et al. S
V Fig. 1. An element of material containingdistributed defects. k k VaoJo.,
(2)
where V~ is the volume of the kth defect, co~ is time-invariant and describes the influence of the shape and orientation of the defect. In the case of aligned parallel defects D o = d* +
- ~ ~%,
(3)
where M
l?u = E I?~.
(4)
k=l
For an incompressible matrix Ogkk= 1.
(5)
From eq. (3) we see that for aligned parallel defects under self-similar expansion, we can use a scalar variable to describe the damage process. We consider the matrix as a homogeneous isotropic hardening material. Its hardening behavior is described by an internal variable ~. The damage effect [eqs (1)-(3)] could be represented by a scalar internal variable ft. The specific free energy is supposed to have the following form (p(E~, 0~,/~)
=
¢p, (e,~) + ¢P2(~, fl),
(6)
where E,~ is the elastic strain, 1 e e ~o~(E eu) = ~C~ktE eE ki
(7)
is the elastic recoverable energy and CUkt the elastic moduli. The plastic potential is F ( Z o, A, B)
= ')"~'eq"q- A "3I- B e ( ~'~*, A ),
(8)
where
1
AB=
0qh O~P2
E* = £um U =
[3~-~/'$--,t ~1/2
Z~ = IF,o - ½Zkk 6 o
(9)
Influence of defect shape on ductile materials
339
0.2
0.1
0
0.2
0.4
0.6
0.8
I
1.0
Fig. 2. Instability strain e~ vs stress-state paramater 7. - - Microcrack damage model.---Microvoid damage model. p = p(fl) is the density, a the Cauchy stress, and Z, A and B are generalized forces corresponding
to e e, ~ and ft. Notice that in eq. (6) we do not assume the separability of ~ and fl, so that A and B are general functions of ~ and ft. This stems from the fact that in many cases g depends also on the hardening behavior of the material, and ~o2(~,/~) = f l (~) +fz(]~)
(10)
will be considered as a special case. With the use o f the first and second t h e r m o d y n a m i c s laws, and the a s s u m p t i o n of n o r m a l dissipativity [9] we get Z~ = C~ktee~ 8F
(3Z~
OF
(
t?g
)
Og)
I+BSj
8F
(11)
51-_~
The law of mass conservation yields
IJ + pDPkk = O,
(12)
31 I
21
1 0
I 0.2
I 0.4
I 0.5.
I 0.8
I
1.0
Fig. 3. Maximum stress E m vs stress-state parameter 7. - - Microcrack damage model. - - - Microvoid damage model.
SUN YI et al.
340 1.6-
1.2- ~'~
,..g
0.8 -
",..,
0.4~
0
~.,..
I 0.2
J 0.4
1 0.5
f 0,8
Fig. 4. Instability value of damage /~c vs stress-state parameter ~/. - - - - Microvoid damage model.
I 1.0 Microcrack damage model.
f r o m eq. (11) tgg
(13)
0 It follows that 1
dp
Bp dfl
1 ~g
1
g ~Y~*
(14)
a* '
w h e r e a* = a*(A, B). S o l v i n g eq. (14) g = go e x p ~-~
B=
a* dp
-
(15)
a'q,
p dlJ
w h e r e go is t a k e n a s a c o n s t a n t , a n d 1 dp q = --___
p d#
(16)
= q(fl).
1000~
75C
b
250
0
I 0.05
I 0,10
I 0.15
I 0.20
Fig. 5. Uniaxial stress-strain curve of an aged duplex stainless steel.
Influence of defect shape on ductile materials
2~/R--10 -
-
R=2
-
~
ii~ 21-I ?
90
4
341
AE2 _ I
(a)
25.0 -
(b) 22.5 20.0
\/VVV
17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0 -2.5 -2.5
I
I
I
I
0
2.5
5.0
7.5
I
I
I
[
I
I
1 0 . 0 12.5 15.0 17.5 20.0 22.5 25.0
Fig. 6. The AE2 notched axisymmetric specimen. (a) Specimen geometry, (b) finite element mesh.
Let AI=A,
Az=B,
el=a,
a2=fl.
(17)
The consistency condition becomes OF Z'J+b-Z 0F Ao=0,
(18)
where i,j = 1-3 and u = 1, 2. From eqs (8), (9) and (11) ~A.e ~
. a2¢p2 OF
(19)
so that = _ 1_ _O_ F . H OZq E~'
(20)
OF OZqh OF H =-OA. c7% Oe,, OA~, "
(21)
where
The rate-type constitutive equation can be written as Y-,~= Lij~lDkl,
(22)
342
SUN YI et al.
where OF OF Cijmn ~ m n Cklpq ~ p q
(23)
Lijkl = Cijkl --
H + a~nmnCmnpq We will discuss some special forms o f ~02. (1) a * = a0 = constant. In this case ao dp B . . . . .
p
thus
d~
f*(fl)
=
Oq~2 ---,
(24)
O8
["
8) =
--J f * ( f l ) dfl + f l (~) = f l (ct) + f : ( f l ) .
(25)
(2) a* = ao - kA. F r o m
e=
~-
(26)
we get the following partial differential equation (27)
P ~-~ + O ~-~ = R,
where P = kq,
Q = I,
(28)
R = -aoq,
The general solution of eq. (28) is
(29)
f ( u , v) = 0 ,
where u, v are Lagrange auxiliary functions u= ~ -
dfl = c~ + k ln p + c~
v = ~02 -
(30)
dfl = q~2 - ~0 In p + c2.
3. CONSTITUTIVE BEHAVIOR (1) D a m a g e variable Theoretically, the choice o f damage variable can be an arbitrary function of p. However, some results o f micromechanical analysis may serve as a guide. F r o m eq. (11) Og Z*' DPq = 211 + 2B 0E* ]~eq
~-2(c02- ½)(B 0g "~21'/2
where
(31)
(32)
~*' = ]E~e;~j, 092 = og~e~O. F o r dilute distribution of defects, it is reasonable to assume that
(33)
B 0~, < 1 , thus
ag
D~k = 2B ~
= DPq
go exp
.
(34)
Influence of defect shape on ductile materials
16.o (a)
16.0
12.0
12.0
8.0
8.0
4.0
343
(a)
4.0
"'"-0.2 "0.4 0.6
1 . 0 ~ - -
0.0
0.0
I 0.0
I 4.0
I 8.0
I 12.0
I 16.0
I
0.0
I
4.0
I 8.0
I 12.0
I 16.0
16.o 2h) 16.0
(b) 12.0
12.0 8.0 g.O 4.0
~1.0
4.0 ""0.9
~3"05.0 7 ~" 0.0
•1 1 .
0.0
[
I 0.0
I
4.0
I
8.0
I
12.0
0.0
I
I
4.0
I
8.0
I
12.0
I
16.0
16.0 Fig. 8. Microcrack damage evolution in AE2 notched specimen. (a) At 30 step loading, (b) at 40 step loading.
Fig. 7. Microvoid damage evolution in AE2 notched specimen. (a) At 30 step loading, (b) at 40 step loading.
Comparison with the microscopic analysis prediction [3, 6] gives D~k = DPq(1 - p)f(Y~*, A ),
(35)
it is suggested that 1
dp
pdfl
1 -p
(36)
(1 - p)lgl
~ 1,
(37)
and eq. (33) becomes so that 1
P = 1 +exp(fl --fl0)
=-po)/~,
flo : l n ( ~ p°
(38)
344
S U N Y I et al. P
A
C) _L
0
ao
-r
bo
A'
W P Fig. 9. The c o m p a c t tension (CT) specimen. W = 50 m m , a 0 = 30 m m , b o = 20 ram.
where Po is the initial density. (2) The hardening law We choose here the power-hardening law A -
-a~ 1+~
(39)
,
where
+ ln( t.
(40)
Employing eq. (39) in eq. (21) gives
. = NOy l u)N(l + k q g u0 k + ~
~-
1
-a
.2dq g -~.
(41)
(3) Material instability We consider an element of material subject to monotonous loading. The matrix material is supposed to be plastically incompressible. The criterion of material instability caused by damage development is given by Drucker's inequality ~-,~jDC:= 22H ~
(42)
The calculation is carried out under axisymmetric loading, with different stress triaxiality E22 = E33 = l / E l l ,
E 0 = 0,
i ¢=j.
(43)
To exhibit the influence of defect shape on damage evolution and macro behavior of materials, we deal with two typical damage mechanisms, microvoid damage and microcrack damage. For microvoid damage
% = ½60,
(44)
090 = ninj,
(45)
while for microcrack damage where n is the unit normal vector perpendicular to the crack surface. In the present calculation, we assume that n coincides with the maximum principal stress direction. The material parameters chosen in the calculation are
Influence of defect shape on ductile materials
345
70 60
50
40
30
Ilii11111
I
,L_L/A
! I Ill I I I I l l l [ | l l I Jllllllill!lllll
[ I I ~
20-
10-
I I I
0-10 -30
I -20
I -10
[ 0
I 10
I 20
L 30
[ 40
I 50
Fig. 10. The finite element mesh for CT specimen.
ay = E
u0 = 0.003,
v = 0.3,
N --- 0.25.
(46)
The damage parameters for the two damage mechanisms are chosen to produce the same E ~ curve in uniaxial tension. For microvoid damage 1 - P0 = 0.03,
a* = 0.64ay
(47)
a* = 2ay.
(48)
and for microcrack damage 1 -
Po =
0.01,
Figure 2 shows the variation of the instability strain along the X~ axis with the stress triaxiality. It is interesting to notice that the difference of the variations of instability strains of the two damage models may reach as much as 100%. What is more surprising is the change of the maximum stress, in which the two models predict qualitatively different results (Fig. 3). This indicates that for notched and cracked specimens, the damage will be localized more seriously in the microcracking model than in the void growth model. Further discussion on this point will be given later. On the other hand, the critical value of the damage variable fic predicted by the two models does not show much difference (Fig. 4). It should be noted that such variation of the stress state may also be found in the same damage mechanism with a different damage hardening coupling effect [eq. (30)]. These results signify that the accurate damage evolution description is not only necessary for local damage prediction, but also for the macroscopic material behavior characterization. 4. FRACTURE P R O C E S S ANALYSIS We apply the present theory to study the fracture process in notched and cracked specimens. The realization and calculation of the present model are carried out on EDFs program ALI-BABA, with the Lagrange large deformation formulation. The material parameters used in the calculation are based on an aged duplex stainless steel [11]. The experimental result of the uniaxial stress-strain relation is shown in Fig. 5. The finite element calculation is focused on the typical damage mechanisms. The initial values of the damage parameters are determined from an AE2 axisymmetric specimen, according to the standard procedure described in ref. [10]. For the void growth mechanism 1 - P0 = 10-3,
ty* = 220 MPa.
(49)
346
S U N YI et al.
Node A
cJ
tm
Failure p "
O
Loading Fig. I 1. The failure of the element in front of the main crack.
For the microcracking process 1 - P0 = 10-6,
o'* = 250 MPa.
(50)
The geometry of the specimen and the mesh used in the calculation are shown in Fig. 6. Figures 7 and 8 show the damage distribution in the notched specimen. It is shown that for the void growth mechanism, the damage begins at the root of the notch, then with the increase of the loading the maximum damage region moves towards the specimen center. Fracture begins in the central region of the specimen. This coincides with the "cup and core" fracture surface observed in experiment. However, for the microcracking process the damage always concentrates at the notch root and the fracture first takes place near the root surface. Such a phenomenon was observed in the notched specimen test of aged duplex stainless steel. Based on the above analysis, we investigate further the influence of defect shape on crack growth behavior in a CT specimen (Fig. 9). The eight-node quadrilateral isoparametric elements are employed near the crack tip and over most parts of the specimen. The mesh used in the calculation is shown in Fig. 10. It should be mentioned that one of the special features of the present model is that the fracture initiation and the crack growth can be simulated without artificial interference. With the increase of loading, material instability takes place at the element in front of the main crack which is mostly damaged, the damage localizes and the element stiffness drops toward zero, following eq. (40), thus forming the crack growth. Here, we define the failure of the element as the sudden increase of the opening displacement of the node just above the crack tip (Fig. 11). Since the damage localization depends on the mesh size, we choose the element length
200
10G
0
I
I
I
I
2
3
Aa (ram) Fig. 12. Comparison of the predicted J - R curves for CT specimen with the experiment. damage model. - - - Microvoid damage model. • Experiment.
Microcrack
Influence of defect shape on ductile materials
347
in front of the crack such as to get the prescribed JI, value. In the present calculation this length is taken as 0.55 mm, which agrees with the experimental result of the aged duplex stainless steel. The predicted J-resistance curves for the two damage mechanisms are shown in Fig. 12. The difference is significant. Such different crack growth behavior might be explained by the difference of the two damage mechanisms in the instability and the damage localization behavior, as previously mentioned. These results emphasize again the importance of the defect shape on damage evolution and macro fracture properties of the material. A comparison is also made between the predicted J - R curves and the experimental result for the aged duplex stainless steel (Fig. 12). It is found that the microcrack damage model could give quite a satisfactory result, which also confirms the prediction of the location of fracture initiation in notched specimens. 5. CONCLUSIONS The characterization of the micro defects and damage development is very important in obtaining the accurate macroscopic material response. The present study reveals that the defect shape has significant influence on damage evolution and fracture behavior of the material. This might be important in engineering design. It should be supplemented that the present model is an idealized one and certain simplifications are made in the calculation. Some factors, e.g. defect rotation and non-similar evolution are not considered. These are left for further research. REFERENCES [1] F. A. McClintock, A criterion for ductile fracture by the growth of holes. J. appl. Mech. 35, 363 371 (1968). [2] J. R. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201-217 (1969). [3] B. Budiansky, J. W. Hutchinson and S. Slutsky, Void growth and collapse in viscous solids, in Mechanics o['Solids (Edited by H. G. Hopkin and M. J. Sewell), pp. 13 45. Pergamon Press (1982). [4] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. Part l--yield criteria and flow rule for porous ductile media. J. Engng Mater. Technol. 99, 2-15 (1977). [5] Sun Yi and Wang Duo, A lower bound approach to the yield loci of porous materials. Acta Mech. Sinica 5, 237 243 (1989). [6] M. Y. He and J. W. Hutchinson, The penny-shaped crack and the plane strain crack in an infinite body of power-law materials. J. appl. Mech. 48, 830-840 (1981). [7] T. Guennouni, Plastic and viscoplastic behavior of cracked materials. Eur. J. Mech. 8A, 491-511 (1989). [8] Q. S. Nguyen and H. D. Bui, Sur les materiaux elastoplastiques a ecrouissage positif et negatif. J. Mecanique 13, 321 (1974). [9] G. Rousselier, Finite deformation constitutive relations including ductile fracture damage, in Three-dimensional Constitutive Equations and Ductile Fracture (Edited by S. Nemat-Nasser), pp. 331 355. North-Holland (1981). [10] G. Rousselier, Ductile fracture models and their potentials in local approach of fracture. Nul. Engng Des, 105, 97 111 (1987). [11] S. Bonnet, et al., Evolution of mechanical properties of various cast duplex stainless steels in relation to metallurgical and aging parameters: an outline of current EDF programs. International Workshop on Intermediate Temperature Embrittlement Process in Duplex Stainless Steels, Oxford (1 2 August 1989). (Received 18 March 1994)