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A continuum damage model for ductile fracture of weld heat affected zone
Enginewing hctwe Mechanics Vol. 40, No. 6, pp. 1075-1082, 15’91 Printed in Great Britain.
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A CONTINUUM DAMAGE MODEL FOR DUCTILE FRACTURE OF WELD HEAT AFFECTED ZONE WANG TIEJUN Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, Shaan Xi, 710049, P.R.C. Abatraet-In this paper, the ductile plastic damage behaviour of weld heat alkcted zone (HAZ) is studied by use of continuum dama mechanics (CDM). Based on a continumn damage variable, D, the effective stress concept and tpf” e thermodynamics, a general continuum damage model for isotropic ductile fracture is derived from a new dissipation potential chosen by the author herein. A comparison between the damage model and experimental results is presented and a good agreement is found. The model is also used to analyse the ductile plastic damage evolution in thermally simulated welding coarse-grained HA2 of a low alloy steel. The elfects of stress triaxiality on plastic damage evolution and on ductile fracture of the coarse-grained HAZ are discussed.
1. INTRODUCTION WELDING represents today the most popular method for joining metals. The inhomogeneities introduced by welding have metallurgical and mechanical properties[ 11.The metallurgical inhomogeneities may result in a local deterioration of fracture toughness and ductility. This degradation in mechanical properties may lead to a decrease of service life of the welded joint. For this reason the mechanical inhomogeneities introduced by welding must be taken into consideration in the structural calculation of weldment. Further studies on the role of residual stresses in the fracture process are also desirable. It is well known that the heat affected zone (HAZ) of a base metal, especially the coarse-grained HAZ (CGHAZ) adjacent to the weld metal, has the lowest toughness[2]. The rupture of welded structures usually initiates from this region. So, the study of damage behaviour of the CGHAZ is very important for the life prediction of welded structures. It is valuable to make clear the damage micromechanism and mechanical deterioration of CGHAZ under loading. Many work8 have been devoted to investigating the damage mechanism and fracture behaviours of CGHAZ through fracture mechanics and other methods[l-7]. In siti observation by the author[l shows that ductile damage of CGHAZ can be indueed by comprehension factors. Void nucleation induced by non-metallic inclusions occurs as a result of the cracking of MnS and/or decohesion at the MnS/matrix interface. Void nucleation induced by the M-A constituent (island Martensite, a carbon-enriched phase) occurs by decohesion at the M-A/matrix interface. Voids can also initiate at prior Austenite grain boundaries. MnS inclusions, especially the larger ones embedded in CGHAZ, have a dominant influence on the damage behaviour of a welded joint. They can accelerate damage initiation and development. Fracture mechanics as an engineering tool has been growing steadily in its applications to practical engineering problems. The application of fracture mechanics to welding is, to a great extent, tied to its ability to predict the fracture behaviour of welded joints. However, this application is neither straightforward nor completely free from drawbacks because a welded joint is a complex, inhomogeneous and anisotropic metallurgical system[l]. Continuum damage mechanics (CDM), originally introduced by Kachanov[S] then Rabotonov[9] and later developed by Lemaitre and others[l(tl2], gives a better understanding of rupture problems in structures by the definition of one or several continuum damage variables representing the degradation of the material. The deterioration processes of the material are described in the framework of continuum mechanics and continuum thermodynamics. The application of CDM gives the engineer a new parameter with which to describe the state of materials. It has been used in many areas: i.e. creep[l3], fatigue[l4], creepfatigue[l5] and ductile fracture[ 16, 17] of metals, concrete[ 181and composite materials[ 191etc. A low cycle fatigue damage model for weld HAZ has been proposed by the author[l4]. In the present paper, CDM will also 1075
1076
WANG
TIEJUN
be used to study the ductile damage behaviour of weld HAZ. A general continuum damage model for ductile fracture of weld HAZ is derived from a new dissipation potential chosen by the author, and is used to analyse the damage evolution of CGHAZ. CGHAZ is very narrow (about 0.2 mm) for a low alloy steel. However, this small region plays a very important role in controlling the damage and fracture properties of the joint. In order to study the damage behaviour of CGHAZ, the weld thermal simulation technique is used to enlarge this small region. The thermal simulator chosen is Gleeble-1500 (Dulfers Scientific, Inc., U.S.A). The material chosen is a low alloy steel and the thermal cycle of CGHAZ is measured during welding process (for more details, see ref. [7]). 2. DERIVATION
OF DUCTILE
DAMAGE MODEL
2.1. Thermodynamic approach Only the brief information needed for the more details, see refs [20,21]. The thermodynamical state of a material elastic deformation c;, temperature T, etc. and mechanisms: plastic deformation ~5, damage isothermal case,
comprehension of the model is introduced here. For element can be described by observable variables: internal variables 8, which reflect various dissipative D, etc. Considering a specific free energy Ip in the
@JJ = ‘p(c& B),
(1)
the dissipated power can be written as
and the elasticity law as
ay =v=aE;i* Introducing
the thermodynamical
(3)
forces B associated with fl, we have B
= --*ay ap
(4)
The existence of a dissipation potential 0 is assumed as a convex scalar function of all flux variables and the state variables acting as parameters, from which the constitutive equations for the evolution may be derived[ 121: @ =WQ;c;,B)
(5)
and
j&-g.
(6)
For ductile fracture, the internal variables /3 are divided into: a hardening variable p defined by P=
s
($“yt$)‘/*dt
a damage variable D defined by
where A and Ati are the overall sectional area and effective resisting area respectively. For the virgin state D = Do 2 0 and for the ruptured state D = DCd 1.
Continuum damage
model for fracture of HA2
The forces B associated with the internal variables p and D are denoted respectively. From eq. (a), we obtain
1077 by R and Y
(9) 2.2. Damage strain energy release rate Y Considering the following thermodynamic
potential !f‘e in the isothermal case,
!P - L a,&G(l
(10)
-D),
e-2p
where aykris the fourth order tensor of elasticity and p is the density. From eqs (3) and (4), we obtain uU= p g
= fa&&(l
(11)
-D)
il and
ay Y=Pz = -~a,&&.
(12)
The density of elastic strain energy W, is defined as W,=
s
dW,=
s
a,,d~;=fa&&(l
-D).
(13)
The expression for Y shows that W ~&!!5.
-y=
1-D
(14)
2dD
The variable - Y can then be considered as the elastic strain energy release rate associated with a unit damage growth. - Y can be calculated as a function of the hydrostatic stress a, = 1/3tr(crU) and the Von Mises equivalent stress ges = (3/2~,s,)‘~ where sU is the stress deviatofll2, 161: 2 -
where
’
=
4:
(15)
D)2
fo
-OfPI =;(1+~)+3(1-2~)
2.
0;4
E and p are Young’s modulus and Poisson’s ratio respectively.
2.3. Derivation of the model We choose a dissipation potential 9 as @=f
Y2
$
( >( > _$
P 2M 1-P
1-a
(16)
PC
where S is a temperature-dependent material constant and a is a damage coefficient which is particularly important in understanding the accumulation of damage. A4 is a hardening constant of the material defined by the Ramberg-Osgood hardening law. The hardening law coupled with damage can be described through strain equivalence and the effective stress concept[l2,15, la]: &=Kp”
in which K is a material constant.
(17)
WANGnl?JuN
1078
From eq. (9), we obtain the damage evolution rate:
6-g
=--
P
Y S
1-a’
( >
pm 1-P
(18)
PC
Substituting eqs (15) and (17) into eq. (18), we obtain the general differential constitutive equation for ductile plastic damage: K2 6 =-
o L!!
2ESJO ues
P
l-o
(19)
or
(20)
In the case of proportional loading, the ratio a,,,/~, can be considered as constant with respect to time. Integrating eq. (20) with initial conditions DIP,, = Do, DIP-, = &
(21)
we obtain
(22) and (23)
where go and pE depend upon the stress triaxiality but it is physically admissible to assume that this dependence is the same for both quantities, and that the ratio p. Jpcdoes not depend upon triaxiality and is equal to its value in the one-dimensional ~~$16,221: PO -=- % PC G
(24)
where c,, and ce are the one-dimensional strains at damage threshold and at failure respectively. For the one-dimensional case, c&r, = l/3 and f(u,,,/a,) = 1. Furthermore, in the range of large deformation, elastic strain may be neglected relative to plastic strain, and &’= C. Then, we obtain from eq. (23) that
From eqs (23), (24) and (25), we obtain (26)
and K2 _md 2ES
D-D L,
(27)
Then, the general differential ductile damage constitutiveequation for more general loading cases may be written by substituting eq. (27) into eq. (20):
(28)
Continuum damage model for fracture of HAZ
A comparison of eqs (22) and (23) gives
Substituting eqs (24) and (26) into eq. (29), we obtain a general integrated evolution law for ductile plastic damage under radial loading: D=D,-
(30)
or (31)
3. IDENTIFICATION OF THE MODEL Identification of such a model consists of the quantitative evaluation of coefficients D,,Do, Q,,ceand tl characteristic of each material at each temperature considered. A convenient method to do this is by use of one-dimensional damage measurement through tension tests, such as the a.c. potential damage measurement technique developed by the author[7,14,23], where the damage D is evaluated as Dc~-~
(32)
P
in which V and p are the potential for the virgin material and the damaged material respectively. Using the plate specimen, tension tests have been carried out on the simulated CGHAZ of a low alloy steel at room temperature. The testing machine chosen was a Universal machine of MTS-880. The a.c. potential system founded includes a lock-in amplifier with isolation transformer (EG&G, U.S.A.) and a power amplifier (B&K, Denmark). The ductile damage evolution vs strain plot for the CGHAZ is shown in Fig. 1. From the experimental results, we obtain D,=O.l885,
D
Do=O.O, .c,=O.O36, q,=O.O, a =0.6.
+
o.l+
+
0
+
+
+
+
+
+
+
+
+
+
++
++
1
2 EN%) Fig. 1. Ductile damage evolution versus strain.
4
1080
D 0.1
Z rP(%) Fig. 2. Comparison of the model and experimental results in the one-dimensional case.
Then, the constitutive equation and evolution law for ductile damage of CGEIAZ is as follows: (33)
(34)
4. APPLICATIONS It is well known that plastic deformation is not the only determinant of the ductile fracture of metals and structures. Plastic deformation is highly dependent upon the state of stresses. The present model gives the relationships between the damage, plastic strain and stress triaxiality. The model will be used to analyse the damage and fracture behaviour of CGHAZ. 4.1. Ductile plastic dumage evolution in CGHAZ For the one-dimensional case, we get from eq. (34) that D
=0.1885[1 -(l-&r].
P(%) Fig. 3. Efkt
of stress triaxiality on damage evolution.
(35)
Continuum damage model for fracture of HAZ
1081
Fig. 4. Effect of stress triaxiality on ductile fracture.
The value of D calculated from eq. (35) is plotted against strain in Fig. 2, which is in good agreement with the experimental results. It is clear that the damage value increases slowly at first, approximately linearly with strain. It increases steeply later, which corresponds to the heavy strain softening and macro-crack growth in CGHAZ. 4.2. Egect of stress triaxiality on the damage evolution in CGHAZ It is clear from eq. (34) that ductile damage is affected by stress and strain state. The effects of stress triaxiality and accumulated plastic strain on the damage evolution in CGHAZ are plotted in Fig. 3. This plot shows that the higher the stress triaxiality, the greater the damage rate will be, which is in good agreement with the numerical and experimental results[24,25]. 4.3. EfSect of stress triaxiality on the ductile fracture of CGHAZ When damage D reaches its critical value D,, the strain reaches its critical value and failure occurs. Returning to the model, pc can be expressed by eq. (26). The values of pf calculated from eq. (26) are plotted vs a,&, in Fig. 4. It is clear from the figure that the higher the stress triaxiality, the lower the accumulated plastic strain to rupture.
5. CONCLUSION A general integrated continuum damage model for ductile fracture has been derived from the dissipation potential chosen by the author in the present paper. A damage coefficient a is introduced in the model, which is particularly important in understanding the accumulation of damage. When a = 1, a linear model can be obtained; when a # 1, a non-linear model can be obtained. The model also shows a clear relationship between damage, stress triaxiality and accumulated plastic strain. The range of validity for the model is limited by the hypothesis of isotropy of damage and plasticity and constant stress triaxiality during loading. In more general cases of loading, the differential model is easy to apply together with plasticity equations coupled with damage. The present model can describe the ductile plastic damage evolution in CGHAZ perfectly. Not only can the model describe the quasistatic development of damage, but also the acceleration of damage evolution. The model can also be used to predict the effects of stress triaxiality on the damage state and failure of CGHAZ. Acknowfedgeme~fs-The author gratefully acknowledges the financial support of the Chinese National Natural Science Foundation. Thanks are also due to Dr. Li Deli and Engineer Li Hua for their kind help during experiments.
REFERENCES [l] L. A. de Vedia. Applications of EPFM to welding: current tmnds. In A&znces in Fracture Research, Pmt. ICF7, Vol. 1, pp. 223-230. Pergamon Press, Oxford (1989).
1082
WANG TIEJUN
[2] M. Kocak et al., Microstructure and weld metal matching eifects on heat affected zone toughness. Symp. 8th Znt.Con5 on Offshore Mechanics madArctic Etwineerine (Edited bv M. M. Salama et al.). Vol. 3. vv. 623-631 (1989). [3] J. H.-Chen e? al., Microfracture beha~our induced by MIA constituent (island iiartensikj insimulated‘welding Heat A&ted Zone of HT80 high strength low alloyed steel. Acfa Metall. 32, 1779-1788 (1984). [4] J. G. Blauel and K.-H. Schwalbe (Ed@, Fracture Mechanics of Welds. Mechanical Engineering Publications, London (1987). [S] J. X. Zhang, Y. W. Shi and M. J. Tu, The fully plastic solutions of various matched weldment for plane stress problem. In Advunces in Fracture Research, Proc. ZCF7, Vol. 1, pp. 541-548. Pergamon Press, oxford (1989). [6] A. Fentaine et al., Prevision of the cleavage fracture properties using a local approach application to the welded joint of a structural C-Mn steel. Nucleur Engng Des. 105,77-81 (1987). [A Lou Zhiwen, Wang Tiejun and Zuo Jianxheng, Studies on the damage evaluation of weld Heat Affected Zone. Acto Sol. Mech. Sinicu (submitted). L. M. Kachanov, Time of the rupture process under creep conditions. Zsv. Akud. Nuuk. SSR, Otd Tekh. Nuuk No. 8, 26-31 (1958). Y. N. Rabotonov, Creep Problems in Strucfurul Members. North-Holland, Amsterdam (1969). D. Krajcinovic, Continuum damage mechanics. Appf. Mech. Rev. 37, l-6 (1984). J. L. Chaboche, Continuum damage mechanics, a tool to describe phenomena before crack initiation. Nucleur lIngng Des. 64, 233-247 (1981). J. Lemaitre, How to use damage mechanics. Nuclear Engng Des. 80, 233-246 (1984). S. Murakami, Notion of continuum damage mechanics and its application to anisotropic creep damage theory. J. Z3gngMuter. Technol. 105,99-105 (1983). Wann Tieiun and Lou Zhiwen. A continuum damaae model for weld Heat Affected Zone under low cycle fatigue loadiig. &gng Fracture Mech.‘37, 825-829 (1990). J. Lemaitre and A. Plumtree, Application of damage concepts to predict creep-fatigue failures. J. Engng Muter. Technol. 101,284-292(1979). J. Lemaitre, A continuum damage mechanics model for ductile fracture. J. Engng Muter. Technol. 107.83-89 (1985). C. L. Chow and June Wang, An anisotropic theory of continuum damage mechanics for ductile fracture. &ng Fructure Mech. 27, 547-558 (1987). J. Marars, A description of micro- and macroscale damage of concrete structures. Engng Frucfure Mech. 26,729737 (1986). A. S. D. Wang and G. K. Haritos (Ed@, Dumuge Mechanics in Composites,AD-Vol. 12. ASME, U.S.A. (1987). P. Germain, Q. S. Nguyen and P. Suquet, Continuum thermodynamics. J. uppf. Mech. SO, 1010-1020 (1983). G. Rousselier, Finite deformation constitutive relations including ductile fracture damage. In Three Dimewiomd ConstitutiveZ&&lionsund Ductile Fructure (Edited by Nemat-Nasser), pp. 331-355. North-Holland, Amsterdam (1981). Wei Hua Tai and Bing Xian Yang, A new micro-damage model for ductile fracture. Engng Fructure Mech. 25,377-384 (1986). Lou Zhiwen and Wang Tiejun, A study of damage evolution of Heat Affected Zone in weld joint. Znr. Conf. on Evuluutionof Muteri& Perfonnunce in Severe Environments(EVALMAT 89). The Iron and Steel Institute of Japan (1989). R. Billardon and L. Moret-Bailly, Fully coupled strain and damage finite element analysis of ductile fracture. Nuclear &iagng Des. U&43-49 (1987). Wei Hua Tai, Plastic damage evolution in low-carbon steels. In Adeunces in Fracture Research, Proc. ZCF7, Vol. 3, pp. 2225-2230. Pergamon Press, Oxford (1989). (Received 28 September 1990)
0013-7944/w
s3.00 + 0.00
PergamonPresspk.
A CONTINUUM DAMAGE MODEL FOR DUCTILE FRACTURE OF WELD HEAT AFFECTED ZONE WANG TIEJUN Department of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, Shaan Xi, 710049, P.R.C. Abatraet-In this paper, the ductile plastic damage behaviour of weld heat alkcted zone (HAZ) is studied by use of continuum dama mechanics (CDM). Based on a continumn damage variable, D, the effective stress concept and tpf” e thermodynamics, a general continuum damage model for isotropic ductile fracture is derived from a new dissipation potential chosen by the author herein. A comparison between the damage model and experimental results is presented and a good agreement is found. The model is also used to analyse the ductile plastic damage evolution in thermally simulated welding coarse-grained HA2 of a low alloy steel. The elfects of stress triaxiality on plastic damage evolution and on ductile fracture of the coarse-grained HAZ are discussed.
1. INTRODUCTION WELDING represents today the most popular method for joining metals. The inhomogeneities introduced by welding have metallurgical and mechanical properties[ 11.The metallurgical inhomogeneities may result in a local deterioration of fracture toughness and ductility. This degradation in mechanical properties may lead to a decrease of service life of the welded joint. For this reason the mechanical inhomogeneities introduced by welding must be taken into consideration in the structural calculation of weldment. Further studies on the role of residual stresses in the fracture process are also desirable. It is well known that the heat affected zone (HAZ) of a base metal, especially the coarse-grained HAZ (CGHAZ) adjacent to the weld metal, has the lowest toughness[2]. The rupture of welded structures usually initiates from this region. So, the study of damage behaviour of the CGHAZ is very important for the life prediction of welded structures. It is valuable to make clear the damage micromechanism and mechanical deterioration of CGHAZ under loading. Many work8 have been devoted to investigating the damage mechanism and fracture behaviours of CGHAZ through fracture mechanics and other methods[l-7]. In siti observation by the author[l shows that ductile damage of CGHAZ can be indueed by comprehension factors. Void nucleation induced by non-metallic inclusions occurs as a result of the cracking of MnS and/or decohesion at the MnS/matrix interface. Void nucleation induced by the M-A constituent (island Martensite, a carbon-enriched phase) occurs by decohesion at the M-A/matrix interface. Voids can also initiate at prior Austenite grain boundaries. MnS inclusions, especially the larger ones embedded in CGHAZ, have a dominant influence on the damage behaviour of a welded joint. They can accelerate damage initiation and development. Fracture mechanics as an engineering tool has been growing steadily in its applications to practical engineering problems. The application of fracture mechanics to welding is, to a great extent, tied to its ability to predict the fracture behaviour of welded joints. However, this application is neither straightforward nor completely free from drawbacks because a welded joint is a complex, inhomogeneous and anisotropic metallurgical system[l]. Continuum damage mechanics (CDM), originally introduced by Kachanov[S] then Rabotonov[9] and later developed by Lemaitre and others[l(tl2], gives a better understanding of rupture problems in structures by the definition of one or several continuum damage variables representing the degradation of the material. The deterioration processes of the material are described in the framework of continuum mechanics and continuum thermodynamics. The application of CDM gives the engineer a new parameter with which to describe the state of materials. It has been used in many areas: i.e. creep[l3], fatigue[l4], creepfatigue[l5] and ductile fracture[ 16, 17] of metals, concrete[ 181and composite materials[ 191etc. A low cycle fatigue damage model for weld HAZ has been proposed by the author[l4]. In the present paper, CDM will also 1075
1076
WANG
TIEJUN
be used to study the ductile damage behaviour of weld HAZ. A general continuum damage model for ductile fracture of weld HAZ is derived from a new dissipation potential chosen by the author, and is used to analyse the damage evolution of CGHAZ. CGHAZ is very narrow (about 0.2 mm) for a low alloy steel. However, this small region plays a very important role in controlling the damage and fracture properties of the joint. In order to study the damage behaviour of CGHAZ, the weld thermal simulation technique is used to enlarge this small region. The thermal simulator chosen is Gleeble-1500 (Dulfers Scientific, Inc., U.S.A). The material chosen is a low alloy steel and the thermal cycle of CGHAZ is measured during welding process (for more details, see ref. [7]). 2. DERIVATION
OF DUCTILE
DAMAGE MODEL
2.1. Thermodynamic approach Only the brief information needed for the more details, see refs [20,21]. The thermodynamical state of a material elastic deformation c;, temperature T, etc. and mechanisms: plastic deformation ~5, damage isothermal case,
comprehension of the model is introduced here. For element can be described by observable variables: internal variables 8, which reflect various dissipative D, etc. Considering a specific free energy Ip in the
@JJ = ‘p(c& B),
(1)
the dissipated power can be written as
and the elasticity law as
ay =v=aE;i* Introducing
the thermodynamical
(3)
forces B associated with fl, we have B
= --*ay ap
(4)
The existence of a dissipation potential 0 is assumed as a convex scalar function of all flux variables and the state variables acting as parameters, from which the constitutive equations for the evolution may be derived[ 121: @ =WQ;c;,B)
(5)
and
j&-g.
(6)
For ductile fracture, the internal variables /3 are divided into: a hardening variable p defined by P=
s
($“yt$)‘/*dt
a damage variable D defined by
where A and Ati are the overall sectional area and effective resisting area respectively. For the virgin state D = Do 2 0 and for the ruptured state D = DCd 1.
Continuum damage
model for fracture of HA2
The forces B associated with the internal variables p and D are denoted respectively. From eq. (a), we obtain
1077 by R and Y
(9) 2.2. Damage strain energy release rate Y Considering the following thermodynamic
potential !f‘e in the isothermal case,
!P - L a,&G(l
(10)
-D),
e-2p
where aykris the fourth order tensor of elasticity and p is the density. From eqs (3) and (4), we obtain uU= p g
= fa&&(l
(11)
-D)
il and
ay Y=Pz = -~a,&&.
(12)
The density of elastic strain energy W, is defined as W,=
s
dW,=
s
a,,d~;=fa&&(l
-D).
(13)
The expression for Y shows that W ~&!!5.
-y=
1-D
(14)
2dD
The variable - Y can then be considered as the elastic strain energy release rate associated with a unit damage growth. - Y can be calculated as a function of the hydrostatic stress a, = 1/3tr(crU) and the Von Mises equivalent stress ges = (3/2~,s,)‘~ where sU is the stress deviatofll2, 161: 2 -
where
’
=
4:
(15)
D)2
fo
-OfPI =;(1+~)+3(1-2~)
2.
0;4
E and p are Young’s modulus and Poisson’s ratio respectively.
2.3. Derivation of the model We choose a dissipation potential 9 as @=f
Y2
$
( >( > _$
P 2M 1-P
1-a
(16)
PC
where S is a temperature-dependent material constant and a is a damage coefficient which is particularly important in understanding the accumulation of damage. A4 is a hardening constant of the material defined by the Ramberg-Osgood hardening law. The hardening law coupled with damage can be described through strain equivalence and the effective stress concept[l2,15, la]: &=Kp”
in which K is a material constant.
(17)
WANGnl?JuN
1078
From eq. (9), we obtain the damage evolution rate:
6-g
=--
P
Y S
1-a’
( >
pm 1-P
(18)
PC
Substituting eqs (15) and (17) into eq. (18), we obtain the general differential constitutive equation for ductile plastic damage: K2 6 =-
o L!!
2ESJO ues
P
l-o
(19)
or
(20)
In the case of proportional loading, the ratio a,,,/~, can be considered as constant with respect to time. Integrating eq. (20) with initial conditions DIP,, = Do, DIP-, = &
(21)
we obtain
(22) and (23)
where go and pE depend upon the stress triaxiality but it is physically admissible to assume that this dependence is the same for both quantities, and that the ratio p. Jpcdoes not depend upon triaxiality and is equal to its value in the one-dimensional ~~$16,221: PO -=- % PC G
(24)
where c,, and ce are the one-dimensional strains at damage threshold and at failure respectively. For the one-dimensional case, c&r, = l/3 and f(u,,,/a,) = 1. Furthermore, in the range of large deformation, elastic strain may be neglected relative to plastic strain, and &’= C. Then, we obtain from eq. (23) that
From eqs (23), (24) and (25), we obtain (26)
and K2 _md 2ES
D-D L,
(27)
Then, the general differential ductile damage constitutiveequation for more general loading cases may be written by substituting eq. (27) into eq. (20):
(28)
Continuum damage model for fracture of HAZ
A comparison of eqs (22) and (23) gives
Substituting eqs (24) and (26) into eq. (29), we obtain a general integrated evolution law for ductile plastic damage under radial loading: D=D,-
(30)
or (31)
3. IDENTIFICATION OF THE MODEL Identification of such a model consists of the quantitative evaluation of coefficients D,,Do, Q,,ceand tl characteristic of each material at each temperature considered. A convenient method to do this is by use of one-dimensional damage measurement through tension tests, such as the a.c. potential damage measurement technique developed by the author[7,14,23], where the damage D is evaluated as Dc~-~
(32)
P
in which V and p are the potential for the virgin material and the damaged material respectively. Using the plate specimen, tension tests have been carried out on the simulated CGHAZ of a low alloy steel at room temperature. The testing machine chosen was a Universal machine of MTS-880. The a.c. potential system founded includes a lock-in amplifier with isolation transformer (EG&G, U.S.A.) and a power amplifier (B&K, Denmark). The ductile damage evolution vs strain plot for the CGHAZ is shown in Fig. 1. From the experimental results, we obtain D,=O.l885,
D
Do=O.O, .c,=O.O36, q,=O.O, a =0.6.
+
o.l+
+
0
+
+
+
+
+
+
+
+
+
+
++
++
1
2 EN%) Fig. 1. Ductile damage evolution versus strain.
4
1080
D 0.1
Z rP(%) Fig. 2. Comparison of the model and experimental results in the one-dimensional case.
Then, the constitutive equation and evolution law for ductile damage of CGEIAZ is as follows: (33)
(34)
4. APPLICATIONS It is well known that plastic deformation is not the only determinant of the ductile fracture of metals and structures. Plastic deformation is highly dependent upon the state of stresses. The present model gives the relationships between the damage, plastic strain and stress triaxiality. The model will be used to analyse the damage and fracture behaviour of CGHAZ. 4.1. Ductile plastic dumage evolution in CGHAZ For the one-dimensional case, we get from eq. (34) that D
=0.1885[1 -(l-&r].
P(%) Fig. 3. Efkt
of stress triaxiality on damage evolution.
(35)
Continuum damage model for fracture of HAZ
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Fig. 4. Effect of stress triaxiality on ductile fracture.
The value of D calculated from eq. (35) is plotted against strain in Fig. 2, which is in good agreement with the experimental results. It is clear that the damage value increases slowly at first, approximately linearly with strain. It increases steeply later, which corresponds to the heavy strain softening and macro-crack growth in CGHAZ. 4.2. Egect of stress triaxiality on the damage evolution in CGHAZ It is clear from eq. (34) that ductile damage is affected by stress and strain state. The effects of stress triaxiality and accumulated plastic strain on the damage evolution in CGHAZ are plotted in Fig. 3. This plot shows that the higher the stress triaxiality, the greater the damage rate will be, which is in good agreement with the numerical and experimental results[24,25]. 4.3. EfSect of stress triaxiality on the ductile fracture of CGHAZ When damage D reaches its critical value D,, the strain reaches its critical value and failure occurs. Returning to the model, pc can be expressed by eq. (26). The values of pf calculated from eq. (26) are plotted vs a,&, in Fig. 4. It is clear from the figure that the higher the stress triaxiality, the lower the accumulated plastic strain to rupture.
5. CONCLUSION A general integrated continuum damage model for ductile fracture has been derived from the dissipation potential chosen by the author in the present paper. A damage coefficient a is introduced in the model, which is particularly important in understanding the accumulation of damage. When a = 1, a linear model can be obtained; when a # 1, a non-linear model can be obtained. The model also shows a clear relationship between damage, stress triaxiality and accumulated plastic strain. The range of validity for the model is limited by the hypothesis of isotropy of damage and plasticity and constant stress triaxiality during loading. In more general cases of loading, the differential model is easy to apply together with plasticity equations coupled with damage. The present model can describe the ductile plastic damage evolution in CGHAZ perfectly. Not only can the model describe the quasistatic development of damage, but also the acceleration of damage evolution. The model can also be used to predict the effects of stress triaxiality on the damage state and failure of CGHAZ. Acknowfedgeme~fs-The author gratefully acknowledges the financial support of the Chinese National Natural Science Foundation. Thanks are also due to Dr. Li Deli and Engineer Li Hua for their kind help during experiments.
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