An analysis of ductile rupture in notched bars

An analysis of ductile rupture in notched bars

J. Mech. Phys. Solids Vol. 32, No. 6, pp. 461 490, 1984. Printed in Great Britain. 0022%5096/‘84 $3.00 + 0.00 , 7;K . (2.11) (TY The increase in ...

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J. Mech. Phys. Solids Vol. 32, No. 6, pp. 461 490, 1984. Printed in Great Britain.

0022%5096/‘84 $3.00 + 0.00 ,



of Engineering,

Brown University, Providence, Rhode Island 02912, U.S.A.

and V. TVERGAARD Department of Solid Mechanics, The Technical


of Denmark,



(Received 13 March 1984) ABSTRACT DUCTILE fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically. based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.



STUDIES of ductile fracture have shown the central role played by microvoid growth in metals (see PUTTICK, 1960; ROGERS, 1960). The voids nucleate mainly at second phase particles, by decohesion of the particle-matrix interface or by particle fracture, and final rupture involves the growth of neighboring voids to coalescence. Analyses by MCCLINTOCK (1968) and RICE and TRACEY (1969) for the growth of a single void in an infinite block of plastic material have shown that void growth is strongly dependent on the level of hydrostatic tension, so that fracture by coalescence of voids is promoted by a high level of triaxial tension. HANCOCK and MACKENZIE (1976) and MACKENZIE et al. (1977) have carried out a series of tests on notched tensile specimens, to study the influence of the stress state on the ductile failure mechanism in high-strength steels. Various circumferential notches on round specimens were used to obtain different degrees of stress triaxiality, as measured by the ratio of the mean tensile stress to the effective Mises stress, and it was 461





found that a higher stress triaxiality results in fracture initiation at a smaller effective plastic strain. More recently, HANCOCK and BROWN (1983) have tested plane strain notched specimens in addition to the axisymmetric specimens, and have found that both types of tests are represented approximately by a single failure locus in a plot of stress triaxiality vs effective plastic strain at fracture initiation. Analyses of the influence of microscopic voids on plastic flow have been carried out on the basis of a set of approximate constitutive relations suggested by GURSON ( 1975. 1977). In this material model the voids are represented in terms of a single parameter. the void volume fraction, and the voids give rise to an apparent dilatancy and pressure sensitivity of the macroscopic plastic deformations. An important effect of the porosity is that even relatively small volume fractions of voids give rise to plastic flow localization at realistic strain levels (see NEEIILI-;:MAN and RICK 1978 ; YAMAMoTo. 1978 ; TVERCGAAKD, 198 1 ; SAJE et ul., 1982). The development of shear bands in a uniformly strained solid containing voids has been studied by SAJE et (11.(1982) for plane strain as well as axisymmetric conditions, under various levels of stress triaxiality. It is found that the material is generally far more resistant to shear band formation when strained under axisymmetric conditions than under plane strain conditions. Thus, a plot ofthe stress triaxiality parameter vs the critical strain for localization will not give a single failure locus valid for both plane strain and axisymmetric deforlnation. However, depending on the stress state. failure may occur prior to localization. and furthermore failure in a single shear band across the specimen may be inhibited by the geometry of some of the notched specimens. A reasonably good agreement with a single failure locus for several different stress states has been found by TVERC;AARD(1982a) in a numerical study of flow localization between larger voids, where the constitutive relations for a porous ductile material have been used to model a population of small scale voids in the matrix material between the larger voids. An extension of the Gurson model to account for final material failure at a realistic value of the void-volume-fraction has been proposed by TWRC;AARD (1982b). Based on experimental coalescence studies (BROWN and EMWRY, 1973; Goons and BROWN, 1979) and numerical model analyses (ANDERSSON,1977) the complete loss of material stress-carrying capacity due to void coalescence is taken to occur at a critical voidvolume-fraction of the order of 0.15 to 0.25. Numerical studies based on this material model have shown completely different modes of final failure in plane strain tension (TVERGAARD. 1982~) and in an axisymmetric tensile specimen (TVERC;AARD and NEEDLEMAN, 1984), as is also observed experimentally (SPWH and SPITZIC;, 1982). Thus, in plane strain tension, localization of plastic flow is predicted at a small voidvolunle-fraction, and subsequently, when the localized deformations grow large, ductile fracture occurs in a void-sheet inside the band. In the round bar tensile specimen, void coalescence in the center of the neck is predicted prior to localization, leading to a macroscopic crack that grows to form the well known cup-cone fracture (Pu~~IcK, 1960; BLUHM and MORRISSEY, 1966). Here, computations for notched tensile specimens, based on the Gurson model, are used to analyze the influence of the stress state on ductile fracture. Some compL]tations of this kind, accounting for the growth of pre-existing voids in round notched specimens, have been carried out by BROWN et al. (1980). In the present paper both

An analysis of ductile rupture in notched bars


axisymmetric and plane strain notched specimens are analyzed. Furthermore, we employ models of both stress and plastic strain controlled void nucleation and our analysis are based on an extension of the GURSON (1975, 1977) model to account for final material failure. Various aspects of the results are compared with the test results of HANCOCK and BROWN (1983). 2.


Relative to a fixed Cartesian frame, the position of a material point in the initial configuration is denoted by r. In the current configuration the position of the material point initially at r is f. The displacement vector u and deformation gradient F are given

Our analysis is based on a convected coordinate formulation of the governing equations (see e.g. GREEN and ZERNA (1968) and BUDIANSKY (1969)) Convected coordinates, xi, are introduced which serve as particle labels. The convected coordinate net can be visualized as being inscribed on the body in the reference state and deforming with the material. The displacement vector and deformation gradient are considered as functions of the xi and a monotonically increasing time-like parameter t. With gi denoting the reference base vectors, the base vectors of the convected coordinate net in the current configuration, gi, are given by & = F!igj


In the reference state the metric tensor is 17ij= gi’gj. In the current tensor is yij = gi * &. The Lagrangian strain increments, qijt are given by ?jij = +(Fy&j+ and written

state the metric



as the sum of an elastic part, pj$ and a plastic part, 45, so that riij zz.z+; + rj&


The elastic part, lie, is obtained from a rate form of Hooke’s law, while the plastic part, qfi, is determined from a flow rule for a progressively cavitating ductile solid based on one introduced by GuRsoN(~~~~, 1977). From an approximate rigid-plastic analysis of a single spherical void, GURSON (1975, 1977) proposed a yield function for a solid with a randomly distributed volume fraction,J of voids of the form (2.5) where cM is the tensile flow strength

t$ = gkjokj,

of the matrix material

0: =


#.i =


and 1

_ o;yij,








with crkj being the contravariant components of the macroscopic Cauchy stress tensor referred to the current convected base vectors. The parameter 4, was introduced by TVERC;AARD(1981, 1982d) who analyzed the macroscopic behavior of a doubly periodic array of voids using a model which fully accounts for the nonuniform stress field around each void and which also accounts for the interaction between neighboring voids. Results for the onset of shear band instabilities obtained from this numerical analysis, for both plane strain and axisymmetric conditions, were compared with results obtained from phenomenological continuum constitutive relations. This comparison suggests y, = 1.5 which is the value we employ. Further modifications of the plastic flow rule, made by TVERC;AARD (1982b) and TVERC;AARDand NEEDLEMAN(1984), are associated with modeling the complete loss of stress-carrying capacity accompanying void coalescence. In particular, the function ,f*(,f) was introduced (TVERCIAARDand NEEDLEMAN. 1984) to model the loss of load-carrying capacity associated with void coalescence occurring at void spacings of the order of the void length, as discussed by BROWN and EMMRURY (1973) and Goons and BROWN (1979). When f* is identified with the void-volume-fraction .f; (2.6) is the yield function proposed by TVERC;AARD(198 1) and if, additionally, y1 is taken to be unity (2.6) reduces to GURSON’S (1975, 1977) original proposal. As in TVERGAARD and NEEDLEMAN(1984) the function f’*(.f’) is taken as,


where f’: = l/q,, .f, is a critical value of the void volume fraction, at which the material stress carrying capacity starts to decay more rapidly and ,fP is the actual void volume fraction associated with the complete loss of stress-carrying capacity. Although the value q1 = 1.5 is chosen based on analyses at relatively small void-volume-fractions, it should be noted that when f* =f,* = l/q, the stress-carrying capacity of the voidmatrix aggregate vanishes. Since the matrix material satisfies the normality rule based on the von Mises yield condition, the macroscopic plastic strain increments satisfy (BERG,1970),

(2.8) The scalar quantity, A, is determined by the strain hardening behavior of the matrix material. The equivalent plastic strain-rate in the matrix is obtained from the plastic work expression and the work hardening properties of the matrix via

a’Gfj = (1 pf)a&P,,


1 -__ E, i

1 . E O”,



where E is Young’s modulus and E, is the current tangent modulus of the matrix material. We take the uniaxial stress-strain behavior of the material to be represented by a

An analysis of ductile rupture in notched bars


piecewise power law of the form


Here CJand E are the true stress and natural strain in uniaxial tension, respectively, and @yis the initial matrix yield strength. Accordingly, E, is given in terms of the matrix flow strength a, by




(> 7;K




The increase in the void-volume-fraction, f; arises partly from the growth of existing voids and partly from the nucleation of new voids, so that we write f = (f)grow~h+ Since the matrix material

is plastically (.&Wtil



incompressible, = (1 -~)~ij~~.


Various nucleation criteria have been formulated within this general phenomenological framework, GURSON (1975, 1977), NEEDLEMANand RICE (1978), which can be described by the simple two parameter relation (2.14) where A gives the dependence of the void nucleation rate on the matrix effective plastic strain increment and B gives the dependence on the rate of increase of hydrostatic stress. The parameters A and B are chosen so that void nucleation follows a normal distribution, as suggested by CHU and NEEDLEMAN (1980). Thus, for nucleation controlled by the plastic strain, the parameters are specified by

where fN is the volume fraction of void nucleating particles, sN is the mean strain for nucleation, and sN is the corresponding standard deviation. The nonzero value of A is only used if& exceeds its current maximum in the increment considered. For materials in which nucleation is controlled by the maximum normal stress on the particle matrix interface, NEEDLEMANand RICE (1978) have suggested using the sum aM + ~$3 as an approximate measure of this normal stress, thus taking A = B. Then an expression analogous with (2.15) is used for A = B. Using the consistency condition C$= 0 together with (2.8), (2.9) and (2.12) to (2.14) gives (2.16)









Plastic yielding initiates when d, = 0 for (i, > 0, and continued plastic loading requires C$= 0 and ~~~~ki~~ > 0. We note that the modi~cation of d, by the function f’* enters (2.17) through the explicit expressions for the derivatives of 4. In order to illustrate the effect ofthe failure terms in the modified Gurson constitutive relation, Figs. 1 and 2 display the response of a homogeneously deformed material element subject to two deformation histories. Figure 1 displays the response to an axisymmetri~ tensile history, while Fig. 2 is for the plane strain tensile history. In each case there is a superposed hydrostatic tension, with the hydrostatic tension history taken to be that experienced by a material element at the center of a necked tensile bar as given by BRIDGMAN’S(1952) formulas. The material parameters used in the numerical calculations described subsequently, namely n,,,jE = 0.0033, 11= 0.3 and n = IO, are also employed here. Furthermore, as in TVERGAARD and NKDLEMAN (1984), we take ji = 0.15 and .fr = 0.25 in (2.7). The choice of these values is based on observations indicating coalescence of neighboring voids when their length is of the order of their spacing, see e.g. Goons and BROWN (1979). These values are also consistent with various analytical estimates, BROWN and EMBIJRY(1973) and ANDERSSON (1977). which indicate void coalescence at void-volume-fractions of the order 0.15 to 0.25. In our calculations we consider the implications of several descriptions of void nucleation for failure. Both plastic strain controlled nucleation and stress controlled nucleation are employed. In each case the volume fraction of void-nucleating particles, J;,, is taken to be 0.04. We carry out calculations with two values of the mean nucleation strain, zN = 0.3 and E,~= 0.8, using the standard deviation sN = 0.1. When stress controlled nucleation is employed we take CJ~= 2.20, and sAv=,. Figure la displays curves of macroscopic effective stress vs macroscopic effective strain for the axisynlmetri~ tension history. For the matrix material this curve follows the power law relation (2.10) and is monotonically increasing. Here, stress controlled nucleation with oN = 2.20, and strain controlled nucleation with c,~ = 0.3 give virtually identical behavior. Around E, = 0.3 the burst of nucleation leads to a true stress maximum. When E,~= 0.8 the onset of nucleation is considerably delayed and so, correspondingly, is the attainment of the true stress maximum. Figure lb shows the development of the void-volume-fraction. As long as the void-volume-fraction, ,I; is less than ,#i.,here equal to 0.15, the failure modification. (2.7), is not active. The rather sharp drop in the stress strain curves in Fig. la is associated with .f’ = .f, = 0.15. When the void-volume-fraction attains the value .fk the stress carrying capacity vanishes. This occurs at c, = 1.17 for the cases where oN = 2.20, and cN = 0.3. and at c, = 1.48 with cN = 0.8. We note that when the modification (2.7) is not employed, i.e. when .f’* = .f; the stress-carrying capacity vanishes for ,f’==2/3, since we use ~1, = 3/2. For this axisymmetric deformation history, with cN = 0.3, the loss of load-carrying capacity with .f* = .f’ occurs at 6, = I .9.

An analysis

of ductile rupture

in notched




0.15 LI 0.10



FIG. 1. Effect of the failure terms in axisymmetric tension, with necking according to Bridgman’s (a) Effective stress vs effective strain. (b) Void volume fraction vs effective strain.


Figure 2 illustrates the response for the plane strain history. Since the Mises effective stress is plotted in Figs. la and 2a any differences between corresponding curves in these figures is due to the effect of stress state on void nucleation and growth. Note that in plane strain the response of the material characterized by stress nucleation with cr,,, = 2.20, differs from that of the material with sN = 0.3. The effective strains at which







1 1.5 Ee FIG. 2. Effect of the failure terms in planz strain tension, with necking according to Bridgman’s (a) Effective stress vs effective strain.(b) Void volume fraction YSeffective strain. 0.5


the stress-carrying capacity vanishes are E, = 0.82, 0.94 and 1.31 for the cases with CJN = 2.2a,, eN = 0.3 and Ed = 0.8, respectively. This dependence of the effective strain at which the stress-carrying capacity vanishes on stress state is expected since the plane strain state gives rise to a higher hydrostatic tension than does the axisymmetri~ state and this promotes increased void growth, MCCLIN-KKK (1968), RKCEand TRACY ( 1969).

An analysis

of ductile rupture

in notched



The present results give a somewhat earlier onset of failure for the plane strain tensile history than for the axisymmetric tensile history, both when failure is identified with the “knee” in the stress-strain curves of Figs. 1 and 2 and when failure is associated with the actual loss of stress-carrying capacity. The earlier failure in plane strain is primarily due to the increased stress triaxiality associated with that stress state, although there is also an effect of stress history. In order to isolate the dependence of failure strain on stress triaxiality, we consider an imposed stress history with constant stress triaxiality. With o$‘~cT, a prescribed constant, incremental quantities in the constitutive relation (2.5) through (2.17), 1 gi, and the macroscopic effective strain rate C,, can be directly expressed in terms of a prescribed effective stress increment de. These relations can then be integrated to obtain the resulting deformation history. A property of the constitutive relation employed here is that the histories of J cM and E, do not depend on the ratios of the individual stress components when 0$30, is maintained constant. As noted above, the failure strain can be identified either as the macroscopic effective strain at which f = f, or as the macroscopic effective strain at which f = fP Curves of failure strain vs stress triaxiality are depicted in Fig. 3 using both these definitions for the case where Ed = 0.3 ; qualitativeIy similar curves can be obtained for the other nucleation criteria employed in this study. Since the failure strain curves in Fig. 3 do not depend on stress component ratios, they pertain to both plane strain and axisymmetric states. The trajectories followed in this parameter space by the deformation histories of Figs. 1 and 2 are plotted in Fig. 3. The points corresponding to f = f, and f = fr are




FIG. 3. Curves of failure strains corresponding to constant values of the stress triaxiality &3/3rr,, for material with plastic strain controlled nu~ieation, EN = 0.3. The two defo~ation histories from Figs. 1 and 2 are shown. The points marked 0 correspond to f = f, and the points marked x correspond to f = &






marked in the figure. In plane strain tension the triaxiality drops rapidly when the void volume fraction exceeds fi. This is due to the increased compressibility of the voided material giving a smaller effective Poisson’s ratio, which relaxes the plane strain constraint. For axisymmetric tension there is no similar effect and the triaxiality continues to rise. As seen in Fig. 3, the failure strain depends on the triaxiality history and also, in general, depends on the ratio of stress components, i.e. on the deformation state. However, the onset of failure along these two non-proportional stressing histories, whether failure is identified with j = fi or .f’ = j;: or some intermediate value of the void volume fraction, is in fairly close proximity to the failure curves obtained for constant triaxiality. This suggests the possibility that there is a range of histories for which a single failure locus may be a reasonable approximation. Subsequently, we will employ a failure curve for constant stress triaxiality as a reference curve when analyzing the influence of stress state on fracture initiation for various notch geometries. We note that the above discussion has been carried out presuming a material element deforming homogeneously up to failure. There is the possibility of flow localization into a shear band. For porous plastic solids, such flow localizations have been studied within a framework, described by RICE (1977) that associates the onset of localization with a material instability, YAMAM~TO(1978), NEEDLEMANand RICE (1978), SAJE et ul. (1982). These analyses give much larger localization strains in axisymmetric deformation states than in plane strain deformation states so that in circumstances where localization precedes, and perhaps triggers, failure, a rather strong effect of deformation state on critical conditions for failure is anticipated.



The finite element analysis is based on a Lagrangian formulation of the field equations with the initial unstressed state taken as reference. The approach employed here uses a convected coordinate formulation of the governing equations. This formulation has been employed extensively in previous finite element analyses, e.g. NEED~~~AN (1972), TVERGAARD(1976) and TVERGAARDet al.(I98 I). The finite element implementation of this formulation, with particular emphasis on issues arising in flow localization problems, has been reviewed by NEEDLEMANand TVERGAARU (1983). In our plane strain analysis we employ a Cartesian coordinate system in the reference state with the x1 -x2 plane being the plane of deofrmation while in the analysis of round bars we use a cylindrical coordinate system with axial coordinate x’, radial coordinate x2 and circumferential angle x3. In the latter case, attention is confined to axisymmetric deformation so that for both geometries ail field quantities are independent of x3. Equilibrium is expressed in terms of the principle of virtual work &jtlijd‘V= sP Here, r’j are the contravariant convected coordinate net. I/and

7%~~ dS.


.s i’ components of Kirchhoff stress on the deformed S are the volume and surface, respectively, of the body

An analysis

in the reference configuration,

of ductile rupture

in notched



and Ti = (r’j + rk&i )n. ,k .I’


where n is the normal to S in the reference configuration. We focus attention on the two notched bar geometries used in the tests of HANCOCK and BROWN (1983). Following the terminology of HANCOCK and BROWN (1983) the more blunt notch is termed the A notch and the sharper notch is referred to as the D notch. These geometries are depicted in Fig. 4 and Fig. 5, respectively. Superimposed on these geometries are typical finite element meshes used in the analyses. Each quadrilateral shown consists of four constant strain “crossed” triangles. The initial length and width of the specimens is denoted by 2L, and 2b,, respectively. In all cases symmetry about the midplane x1 = 0 is assumed. Furthermore, in plane strain, symmetry about the line x2 = 0 is enforced so that, for both the plane strain and axisymmetric cases, only one quarter of the meshes shown in Figs. 4 and 5 is analyzed numerically. A uniform end displacement, U, is prescribed in the x1 direction and the boundary take the form conditions for the quadrant analyzed numerically T2 = 0,

T2 = 0,




T’ = 0,


x1=0 T2 = 0

x1 = L 0, for on

(3.3) b

0 6 x2 < 2, 2 S,,

(3.4) (3.5)

where SL is the lateral surface of the specimen. At a given stage of the imposed loading history the current values of all field quantities, e.g. stresses, r’j, strains, rij, void volume fraction, S, are presumed known. An increment of end displacement, U, is prescribed. Expanding the principle of virtual work (3.2) about the current known state gives to lowest order y{?jR~ij+

~~~~~~ ” ,,6 u,.j)d1/=

b f’bilidS_[




s The bracketed terms vanish if the given state is precisely an equilibrium state. However, due to numerical errors arising from a finite step size, the given state may not exactly satisfy equilibrium. The bracketed correction terms then compensate for this lack of equilibrium and inhibit the accumulation of numerical errors arising from the incrementation scheme. Numerically, the material failure incorporated into the constitutive relation discussed in Section 2 is implemented via the element vanish technique of TVERGAARD (1982~) which was also employed by TVERGAARD and NEEDLEMAN(1984). When the failure condition is met in an element (i.e. when f* = f:) the element is taken to vanish in that it no longer contributesto the virtual work integral (3.2) but the nodal points associated with vanished elements are not removed. In order to avoid numerical difficulties associated with nearly vanished elements, the elements are actually taken to vanish when f* = 0.9 f,*, and the remaining small residual nodal forces are released gradually in subsequent increments.






Frc. 4. A-notch


with a typical mesh. h,;‘&, = 0.333. R,/h,,

= 0.5.


FIG. 5. D-notch

0.333, R,/b,,

= 0.167.

An analysis

of ductile rupture


in notched




The two different notch geometries shown in Figs. 4 and 5 are analyzed both for plane strain specimens and for axisymmetric specimens, and these investigations are carried out for three different material models. Two of these materials have plastic strain controlled nucleation, with mean nucleation strains Ed = 0.3 and Ed = 0.8, respectively, and with the standard deviation sN = 0.1. The third material has stress controlled nucleation, with crN = 2.20, and sN = 0.1~~. The remaining material parameters o,/E = 0.0033, v = 0.3, n = 10, q1 = 1.5, fN = 0.04, S, = 0.15 and fF = 0.25 are common for all three materials, and the initial void volume fraction is taken to be zero in all cases. It is noted that the material with Ed = 0.3 is identical with that investigated by TVERCAARD (1982~) for uniaxial plane strain tension and by TVERCAARD and NEEDLEMAN(1984) for axisymmetric tension. In uniaxial plane strain tension localization of plastic flow in shear bands occurs at the logarithmic strain E~ = 0.23 for this material, and subsequently ductile fracture occurs by void coalescence inside the bands. The analysis of the round tensile bar showed initial void coalescence in the center of the neck, prior to shear band formation, leading to a cup-cone fracture. The calculated load vs average axial strain curves for the notched specimens are shown in Figs. 6-9. Results are given for the plane strain A-notch in Fig. 6, for the plane strain D-notch in Fig. 7, for the axisymmetric A-notch in Fig. 8, and for the axisymmetric D-notch in Fig. 9. Curves for all three materials are given in each figure, except for the axisymmetric A-notch, which has not been analyzed with Ed = 0.8.




o.ot’,““‘,“,“,‘,“, 0





E. FIG. 6. Load vs average axial strain curves for plane strain A-notch.









0.0 0





% FIG. 7. Load vs average

axial strain wwes

for plane strain D-notch.



0.0 0





Ea FIG. 8. Load vs average

axial cuwes for axisymmetric



An analysis of ductile rupture in notched bars 0.6




I / , ‘1 I

I ,

I 1 I ,



S 0.2

FIG. 9. Load vs average axial strain curves for axisymmetric D-notch.

before nucleation occurs, the three curves coincide; but soon after nucleation has taken place the slope of the curve starts to decay rapidly. For each of the four geometries analyzed it is seen that the material with gN = 2.20, gives the earliest nucleation, the material with sN = 0.3 is next and that with E,,,= 0.8 is last. It is noted that the average strains are rather small in all four figures, because large strains, as those shown in Figs. 1 and 2, occur only in the vicinity of the notches. In the following, Figs. l&18, we focus on the behavior of the material with cN = 0.3. Figure 10 shows contours of constant void volume fraction at three stages of deformation in a plane strain A-notch. It is seen that nucleation first occurs in a rather smooth region at the notch surface, where the strains are relatively large; but subsequently localization of plastic flow takes place, and Fig. 1Oc shows that failure is going to occur in a shear band, starting at the center. Fig. 11 gives contours of constant maximum principal logarithmic strain in the plane strain A-notch at the same three stages. The initial strain concentration at the notch surface and the subsequent localization in a shear band are clearly seen, and it is noted that the strains outside the notch region remain small. The mesh used for the analysis is designed so that the diagonals are inclined nearly 40”, when critical strains for localization are approached. Previous investigations have shown that representing localization with reasonable accuracy requires careful mesh design (TVERGAARD et al., 1981; TVERGAARD, 1982~). The development of the stress triaxiality, c$3~7,, in the plane strain A-notch is illustrated in Fig. 12. The stress triaxiality is maximum at the center of the specimen in the first two stages of deformation shown in this figure. At the later stage of Initially,



0 001 001

0001 I

00’ ( b)


FIG. 10. Curves

of constant

void volume fraction for plane strain A-notch (a) E, = 0.0194, (b) E, = 0.0269. (c) c, = 0.0288.

0 025

0 05




! I’


with E,~= 0.3

0 025


?‘O _

( b) FIG. 11. Curves ofconstant

and material


maximum principal logarithmic strain for plane strain A-notch and material I:,? = 0.3. (a) I:, = 0.0194, (b) c,, = 0.0269, (c) I:, = 0.028X.


An analysis

(0) FE.

12. Curves

of constant

of ductile rupture

in notched



( b)

triaxiality &3cr, for plane strain A-notch (a) S* = 0.0194, (b) r, = 0.0269. (c) 8, = 0.0288.

CC) and



cN = 0.3.

deformation, when fracture has initiated in the center of the specimen (f > f,, in Fig. lOc), the peak triaxiahty has shifted upward from the specimen center. For the same material, sN = 0.3, void volume fraction contours at a plane strain Dnotch are shown in Fig. 13. The void volume fraction distribution around this sharper notch differs significantly from the distribution shown in Fig. 10. In both cases nucleation starts at the notch surface; but in Fig. 13 the initial fracture site is at the surface too, while there are nearly no voids at the center of the notched section. The pattern shown in Fig. 10 has still some similarity with the behavior of an unnotched specimen, whereas Fig. 13 approaches the situation at the tip of a sharp crack. The principal strain contours in Fig. 14 show that the peak strains occur at the notch surface and propagate into the specimen in a rather well defined shear band. This contrasts with Fig. 11, where it can be seen that for the more blunt A-notch, the peak strains occur at the specimen center. Figure 15 shows the stress triaxiality 0$3a, at the plane strain D-notch. The triaxiality at the center of the notched section is high; but, as can be seen in Fig. 13, voids hardly grow in this region of high triaxiality, because the strains, Fig. 14, are not large enough to cause nucleation. A much finer mesh than that shown in Fig. 5 has been used to study the initial growth of the crack from the surface of the plane strain D-notch. For numerical convenience b$L., was taken as 0.5 in this calculation. Part of this mesh, in the notch region, is shown in Fig. 16 at three different stages of the deformation. As mentioned in Section 3 the elements are taken to vanish when j’ = fF, and all vanished (fractured) triangular



I . ,*,* IOOOI~ 001

0005 005

FIG. 13. Curves

of constant





1, 0 I5


void volume fraction for plane strain D-notch (a) c,, = 0.0272, (b) i:,, = 0.0390.


and material

with i:, = 0.2.

CO25 0025 005 010 ~



305 ~'0025

0 IO


',_ 040

(01 FIG.14. Curves of constant

,' 0'05


maximum principal logarithmic strain for plane strain D-notch and material 8N = 0.3. (a) 8, = 0.0272 and (b) t:,, = 0.0390.


An analysis

of ductile rupture

of constant




(a) FIG. 15. Curves

in notched

triaxiality ~:/3a, for plane strain D-notch (a) a. = 0.0272, (b) E, = 0.0390.




eN = 0.3.


FIG. 16. Crack growth in the notch region for plane strain D-notch and material with aN = 0.3. Vanished triangular elements are painted black. For this calculation b,/L, in Fig. 5 is 0.5 and between (a) and (c) E, varies between 0.051 and 0.053.






elements are painted black at the three stages shown in Fig. 16. It is seen that the crack keeps growing straight ahead, even though the void distribution in Fig. 13 seems to suggest that the crack might grow in a direction inclined about 45” with the central cross section. It is noted that this method of analyzing crack growth by void coalescence in a ductile material was also employed by TVERGAARD (1982~) and by TVERGAARD and NEEDLEMAN(1984). Figure 17 shows contours of constant void volume fraction in an axisymmetric Anotch specimen. The greatest void nucleation and growth occurs at the center of the specimen. The region of high void volume fraction propagates from the specimen center toward the surface, remaining quite close to the minimum section. This behavior is quite similar to that found for an unnotched axisymmetric tensile specimen, TVERCAARD and NEEDLEMAN(1984). For an axisymmetric D-notch, void volume fraction contours are shown in Fig. 1X, still corresponding to cN = 0.3. It is clear that initially the voids nucleate and grow significantly at the notch surface; but subsequently the rapid void growth due to the high triaxiality at the center takes over, and fracture initiates at the center. Between the surface region and the central cracked region there is a region with a relatively low void volume fraction in Fig. 18b. The material with a high mean strain for nucleation, cN = 0.8, has been analyzed for two plane strain notches and one axisymmetric notch. As an example, Fig. 19 shows results at a rather late stage of the plane strain A-notch calculation. Comparing with the initial geometry (Fig. 4), it is seen that there is a considerable thinning of the



0001 \



002 (01 FIG. 17. Curves of constant




00; Cb)

1 0 IO

void volume fraction for axisymmetric A-notch (a) i:, = 0.0187, (b) I:, = 0.0224.

and material

with I:,~= 0.3.

An analysis

of ductile rupture

in notched



303 125 j

FIG. 18. Curves

of constant

void volume fraction for axisymmetric D-notch (a) E, = 0.0137, (b) E, = 0.0158.

and material

with ~~ = 0.3.




I‘10 6


~-, -.

0025 //’


\ ‘,






FIG. 19. Plane strain A-notch and material with cN = 0.8, at E, = 0.048. (a) Curves of constant fraction, (b) maximum principal logarithmic strain, (c) triaxiality &30,.

void volume




specimen in the notched region, because the strains have to grow large before the voids are nucleated. These large strains are seen on the contours of maximum principal logarithmic strain in Fig. 19b. Otherwise the pattern of void growth (Fig. 19a) and the distribution of stress triaxiality (Fig. 19~) are in rather good agreement with the results shown in Figs. 10 and 12. It is noted that fracture initiates in the center of the specimen. Also in the analyses of the plane strain D-notch and the axisymmetric D-notch with c,~ = 0.8, the behavior found is analogous with the results of Figs. 13, 14, 15 and IX, when account is taken of the larger strains required for nucleation. The next three Figures, 20-22, show results for the material with stress controlled nucleation, gN = 2.20,. Void volume fraction contours in the plane strain A-notch are shown in Fig. 20. High stresses are reached at an early stage of the process, and therefore a relatively high void volume fraction develops in a larger part of the notch region than found in the case of Fig. IO. The curves in Fig. 2Oc indicate that also hcrc failure is going to occur in a shear band, starting in the center. An interesting feature of the nucleation law is seen in Fig. 20a, where interaction between relatively high values of the matrix flow strength CT,,,at the notch surface and relatively high triaxial tension ~$3 at the center gives the strongest initial nucleation between these two points. Results for the plane strain D-notch, with stress controlled nucleation, are shown in Fig. 21. Also here the porosity (Fig. 21a) is higher over a larger part of the notch region than found in Fig. 13 ; but fracture still initiates at the notch surface. The distribution of the triaxiality (Fig. 21~) is rather similar to that found in Fig. 15. For the axisymmetric D-notch with stress-controlled nucleation, contours of void-

0 001 001 0 001


0 01


03: /



10 IO


FIG. 20. Curves of constant



void volume fraction for plane strain A-notch and material (a) E, = 0.00408, (b) E, = 0.00668, (c) c,, = 0.0117.

with ci\, = 2.20,.

An analysis

of ductile rupture

in notched






FIG. 21. Plane strain D-notch and material with cN = 2.2c,, at E, = 0.0175. (a) Curves of constant volume fraction, (b) maximum principal logarithmic strain, (c) triaxiahty &3a,.


QEL__ - -001 -, @O~_T<, 0075 010



010 (b)



FIG. 22. Axisymmetric D-notch and material with oN = 2 2 oy,at E, = 0.00589. (a) Curves of constant volume fraction, (b) maximum principal logarithmic strain, (c) triaxiality ~$30,.





volume-fraction, of maximum principal strain, and of triaxiality are shown in Fig. 22. The voidage is here spread over a smaller region than shown for plane strain in Fig. 2 la, which agrees with the differences between the plane strain and axisymmetric D-notches with strain-controlled nucleation shown in Figs. 13 and 18. As in Fig. 18 fracture initiates at the center, where high triaxiality gives rapid void growth. All of our computations for the axisymmetric specimens exhibit fracture initiation at the specimen center, in agreement with the experimental observations of HANCOCK and BROWN (1983) for high-strength steels. Also in agreement with these experiments. failure initiates at the surface in plane strain specimens with the sharp D-notch. For the plane strain A-notches most of the experiments of HANCOCK and BROWN (1983) show fracture initiation at the center, in agreement with the present computations; but some experiments show failure initiation both at the center and at the surface. This possibility of simultaneous failure at the two sites agrees with some trends of the void volume fraction contours in Figs. 10, 19 and 20. Particularly Fig. 10 shows that the shear band gives a simultaneous intensification of porosity at the center and at the surface.



The conditions at failure initiation found in twelve different computations are summarized in Table 1. In the computations, the conditions at failure are identified with those prevailing in the element with the highest void volume fraction,.f; just prior tof’ = ,f,. In all casesf, is taken as 0.15. Whenfattains the valuef, there is a rapid local drop in stress carrying capacity. As discussed by TVERGAARD and NEEDLEMAN (1984) and also found here this is associated with a sharp drop in load. The effective strain



Notch configuration A-notch A-notch A-notch

and macroscopic


Nucleation criterion

strain atfailure

initiation site



Location of failure initiation

plane strain plane strain plane strain

CT&TV= 2.2 EN = 0.3 F,N = 0.8

1.6 1.3 2.1

0.18 0.38 0.89

Center Center Center

D-notch plane strain D-notch plane strain D-notch plane strain

UNJ(TL. = 2.2 F,v = 0.3 l:,v = 0.8

0.71 0.69 0.73

0.48 0.78 1.17

Surface Surface Surface

A-notch A-notch

axisymmetric axisymmetric

ff& = 2.2 EN = 0.3

1.7 1.8

0.23 0.4 1

Center Center

D-notch D-notch D-notch

axisymmetric axisymmetric axisymmetric

u& = 2.2 cN = 0.3 sN = 0.8

2.1 2.4 2.2

0.10 0.25 0.82

Center Center Center

E,~= 0.8






An analysis of ductilerupture in notched bars


shown in Table 1 is a total effective strain calculated from the known principal strains at failure initiation. For the type of strain histories encountered in the present computations this strain agrees well with one based on an actual accumulated plastic strain. In addition to the notched bar results, a result is included in Table 1 for an unnotched axisymmetric tensile specimen with sN = 0.8. In the experimental studies of HANCOCKand MACKENZIE(1976) and HANCOCKand BROWN (1983), local failure conditions were inferred from calculations based on the Mises flow rule for a plastically incompressible solid. Experimentally, fracture initiation was identified with a drop in average true axial stress. The use of this criterion is related to local failure by HANCOCK and MACKENZIE’S (1976) observation that specimens examined close to the drop in stress exhibited a link-up of voids forming a crack, whereas in specimens examined before the load drop only discrete voids were present. As noted above this association is also consistent with the results of the present calculations. With the onset of failure defined in this way, HANCOCKand MACKENZIE (1976) and HANCOCKand BROWN(1983) find that a variety of test results are rather well represented by a single failure locus in a plot of stress triaxiality vs effective strain at fracture initiation. In Section 3 we have mentioned that for the constitutive relation employed here, when the triaxiality ai/3a, is prescribed to be constant, there is a single failure locus independent of the ratios of the individual stress components. Figure 3 shows such curves corresponding to f = f,and f = fFfor the material with sN = 0.3 and gives a comparison with two histories for which the stress triaxiality is not constant. We show plots of stress triaxiality vs effective strain at fracture in Figs. 23, 24 and 2.5 for the





FIG. 23. Stress triaxiality

vs effective strain at fracture for material and reference curve.

with EH = 0.3. Failure points from Table




b” rr) \ Xl b

FIG. 24. Stress triaxiality









vs effective strain at fracture for material and reference curve.

with F~ = 0.8. Failure points from Table


FIG. 25. Stress triaxiality

vs effective strain at fracture for material with uN = 2.2~7,. Failure points from Table I and reference curve.


An analysis of ductile rupture in notched bars


notched bar calculations with sN = 0.3, Ed = 0.8 and cN = 2.2a,, respectively. In each figure the curve for f = fc corresponding to constant values of akJ3o, is shown as a reference, and the circles denote the failure points taken from the values given in Table 1. Since at the material point where fracture initiates, the stress triaxiality, 4/3a,, does not, in general, remain constant throughout the deformation history, some history dependence of the failure strain is expected. In Figs. 23 to 25 it can be seen that there is some scatter around the reference curves, which is a consequence of the history dependence of the fracture strain. The points for the plane strain D-notch specimens, which fail at the surface at a low value of stress triaxiality, all lie close to the failure curve for proportional triaxiality. For these specimens, the deviation from constant stress triaxiality is slight prior to failure initiation. However, examination of Table 1 reveals a certain systematic trend in the deviation from a history-independent failure locus. Failure initiation in the plane strain A-notch specimens occurs with lower triaxiality and at a smaller value of effective strain than for the corresponding axisymmetric A-notch specimens. This behavior of the plane strain A-notch specimens is an effect of shear localization. A smaller fracture strain is expected in cases where failure involves localization and, in general, plane strain states are more susceptible to localization than axisymmetric deformation states. Previous studies of localization in porous plastic solids have found that localization in plane strain typically occurs at relatively small void volume fractions (NEEDLEMAN and RICE, 1978 ; TVERGAARD, 1981; SAJE et al., 1982). However, it is important to note that the critical strains given by such analyses refer to the strain level outside a shear band, where as the strains given in Table 1 are those at the actual failure point which may be inside a band. Full calculations of shear band development have shown that the localized strains required for coalescence inside a band are large (TVERGAARD, 1982b, c). Thus, for the onset of failure as defined here, localization can precede failure and lead to a strong deviation from proportional loading at the eventual failure site. Our calculations indicate that, as long as deviations from a constant triaxiality history are not too great, the onset of failure, when represented in a plot of stress triaxiality vs effective strain, is approximately represented by a single curve. However, significant deviations from this behavior can occur for nonproportional histories, with plastic flow localization playing an important role in this regard. In fact, in a very recent study, THOMSON and HANCOCK (1983) comment that the unique failure locus is meant to apply only to deformation histories with small changes in triaxiality. We find that in plane strain bluntly notched specimens, the nonproportionality arising from shear localization does lead to significantly earlier failure strains. Another important issue concerns the extent to which an analysis using the classical, plastically incompressible, Mises flow rule gives stress and strain values representative of local conditions in a voided solid. Void nucleation and growth leads to a lowering of local stress levels due to the softening that develops with increasing porosity, as illustrated in Figs. 1 and 2. This role of porosity in lowering stress levels was also noted by BROWN et al. (1980), who carried out calculations for axisymmetric A-notch and Dnotch specimens with various initial porosities, using GURSON’S (1975, 1977) constitutive relation. An additional important effect that we find is the loss in triaxiality in plane strain associated with increasing porosity. This is already illustrated in Fig. 3, where a






drop occurs during the failure process f > f,. In the nonuniform deformation fields that occur in the notched bar analyses, this effect can take place more gradually over the deformation history and also leads to a significant redistribution of stresses. This effect is clearly illustrated in Fig. 12 where the peak triaxiality has moved completely away from the minimum section. Even for an axisymmetric specimen, there is a significant stress redistribution once failure initiates, Fig. 22. In contrast, for the unmodified GURSON (1975, 1977) flow rule, BROWN et al. (1980) comment that the triaxiality level, i.e. the ratio &3a,, is not much affected by porosity. It should also be emphasized that local stress and strain values obtained from an analysis based on the Mises flow rule cannot reflect the effects of plastic flow localization. Especially for plane strain specimens with blunt notches, stress triaxiality levels inferred from calculations for an incompressible Mises solid may not be representative of local conditions in a voided solid at failure initiation. On the other hand, for more sharply notched plane strain specimens and for the axisymmetric specimens, an analysis based on the Mises flow rule would predict triaxiality levels at failure initiation that differ little from the ones we obtain. Although, as mentioned above, once failure initiates the differences do become substantial. We find that the development of failure is quite sensitive to specimen geometry and deformation state and also depends on the type of void nucleation criterion. For example, for the plane strain D-notch specimen, with both strain controlled and stress controlled nucleation, failure initiates at the notch root even though the stress level is relatively low there. At a later stage there is a difference in response between these two cases. Comparing Fig. 21a with 13b shows that the specimen with stress controlled nucleation has a more extensive region of high void volume fraction, as expected from the fact that high stresses prevail over a larger region than do high strains. However, the overall patterns of void volume fraction in these two figures are broadly similar. Also, all the plane strain A-notch specimens exhibit shear band development, as illustrated in Figs. 10-12 and 19 and, for all axisymmetric specimens, failure initiates in the center. However, here too there are differences in detail between the stress nucleation and strain nucleation cases, compare Figs. 18b and 22a. There may be circumstances where void nucleation effects play a more dominant role in setting the failure mode. In a previous study of failure in axisymmetric tension, TVERGAARDand NEEDLEMAN (19X4), the crack,in its initial stages of growth was confined to the plane of minimum cross section due to a kinematic constraint imposed by the axisymmetric geometry. In Fig. 16, it is not such a geometric constraint, but rather the strong gradients associated with the sharp notch geometry that act to confine crack growth to the midsection plane. This calculation was only carried out for the initial stages of crack growth and even though a relatively fine mesh was employed, the extent to which the behavior is meshdependent cannot be completely specified. Based on previous studies, TVERC~AARD (1982~) and TVERGAARDand NEEDLEMAN(1984) it is expected that the main features of the failure process in Fig. 16 represent actual behavior of the material model and are not artifacts of the discretization. Several features of the failure process in Fig. 16 are noteworthy. First, we emphasize that this failure process is a natural outcome of the constitutive description we employ ; no additional fracture criterion is imposed. Failure does not initiate at the notch surface but slightly below. This is actually expected since the stress-free condition on the notch

An analysis of ductile rupture in notched bars


surface lowers the surface stress levels. Once initiated, the failure region propagates both backward toward the notch and forward into the material. Each subsequent failed element is somewhat removed from the newly created free surface. The failed region remains along the row of elements nearest the midsection despite the clear tendency to shear banding in Figs. 13 and 14. This is of interest in relation to the study of MCMEEKING (1977), who considered a single void growing near a crack tip and found the greatest growth rate at 45 degrees to the crack tip during the initial stages of blunting, whereas straight ahead crack growth was favored at larger crack opening displacements. MCMEEKING (1977) suggested that strain controlled nucleation would favor crack growth at an angle to the crack line while stress controlled nucleation would favor straight ahead growth. Thus, it is worth noting that strain controlled nucleation was employed in the case shown in Fig. 16. Although our results are suggestive for crack behavior, it should be emphasized that a macroscopic notch has been considered here and there are significant differences between factors governing the behavior at such a notch and those pertinent to crack tip behavior. For example, there is an important effect of size scale. The model we have used presumes that gradients of stress and strain fields are small over size scales comparable to void sizes and spacings, while at a crack tip gradients are large over microstructurally significant size scales. Although a detailed quantitative comparison with experiment has not been made here, the present calculation appear to provide accurate descriptions of the failure process in notched specimens and of the effects of stress and deformation state. Previous analyses using a porous plastic continuum model have reproduced the observed failure behaviors of unnotched plane strain and axisymmetric tensile specimens in remarkable detail, TVERGAARD(1982~) and TVERGAARD and NEEDLEMAN (1984). It seems clear that studies ofductile failure carried out within this framework are capable of playing an important role in providing a bridge between continuum descriptions of stress and deformation fields and microstructural descriptions of processes of ductile rupture. ACKNOWLEDGEMENTS A.N. gratefully acknowledges the support of the U.S. National Science Foundation (Solid Mechanics Program) through grant MEA-8101948 and the support of AR0 Grant DAAG2981-K-0121. V.T. is grateful for support provided by the U.S. National Science Foundation (Solid Mechanics Program) through grant MEA-8101948, which made possible a stay at Brown University as a Visiting Professor during which time this investigation began. The computations reported on here were carried out on a VAX-l l/780 computer at the Brown University, Division of Engineering, Computational Mechanics Computer Facility. This facility was made possible by grants from the U.S. National Science Foundation (Grant ENG78-19378), the General Electric Foundation and the Digital Equipment Corporation. REFERENCES ANDERSSON, H. BERG,C. A.

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1960 1977


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