Tvergaard MM 1987

Mechanics of Materials 6 (1987) 53-69 53 North-Holland

DUCTILE SHEAR FRACTURE AT THE SURFACE OF A BENT SPECIMEN

Viggo TVERGAARD

Department of Solid Mechanics, The Technical Universi(v of Denmark, Lyngby, Denmark

Received 20 October 1986

The development of shear bands at the stretched surface of a bent plate is analysed numerically, based on an approximate continuum model of a ductile porous material. This material model accounts for the nucleation and growth of voids as well as the effect of the yield surface curvature, which is represented by a combination of kinematic hardening and isotropic hardening. An imperfection in the form of an initial surface waviness is assumed, which triggers shear bands at the wave bottoms. The corresponding periodic pattern of shear bands is considered, and the growth of the bands is followed, until shear cracks develop from the void-sheets inside the bands. The delay of localization due to the nonuniform strain field is studied for different versions of the material model. Furthermore, the stability of the uniform growth of several adjacent shear bands is investigated.

1. Introduction

Several analyses have shown that microscopic voids in a ductile metal have a significant effect on the localization of plastic flow in narrow shear bands. Most of these investigations have been based on approximate constitutive relations for porous materials (see Needleman and Rice, 1978; Yamamoto, 1978; Saje, Pan and Needleman, 1982), but the same effect of porosity has also been found in a study that accounts for a discrete void distribution (Tvergaard, 1981).

In structural alloys voids nucleate mainly at second phase particles, by particle fracture or by decohesion of the particle-matrix interface (see Puttick, 1959; Goods and Brown, 1979), and subsequently the voids grow due to plastic straining of the surrounding material. Ductile fracture by coalescence occurs when the ligaments between adjacent voids have thinned down sufficiently. A material model that incorporates these features of a progressively cavitating solid has been suggested by Gurson (1977a, b), in which the voids are represented in terms of one scalar damage parameter, the void volume fraction. In this model the growth of the voids gives rise to an apparent dilatancy and pressure sensitivity of the macroscopic plastic

deformations, and nucleation can give rise to a significant non-normality of the plastic flow rule.

Other sets of constitutive relations for ductile porous materials have been developed to describe the deformations of powder metallurgy materials (e.g. Shima and Oyane, 1976). These models do not account for nucleation, but it is found that the yield surfaces used to represent experimental results for powder compacted metals are in reasonable agreement with the modified version of Gur- son's yield surface that has been the basis of a number of ductile fracture studies (see discussion by Tvergaard, 1987).

Recently, Mear and Hutchinson (1985) have extended Gurson's material model by introducing a family of dilatant plasticity theories, in which the yield surfaces change by a combination of isotropic expansion and kinematic translation. Tvergaard (1987) has further extended this kinematic hardening model to incorporate the effect of void nucleation. These constitutive relations make it possible to study the combined effect of porosity and an increased yield surface curvature at the point of loading, and it has been found that localization occurs earlier in the kinematic hardening solid (Mear and Hutchinson, 1985; Becker and Needleman, 1985; Tvergaard, 1987). Previous lo-

0167-6636/87/$3.50 ~ 1987, Elsevier Science Publishers B.V. (North-Holland)

54 V. Tver~;aard / Ductile shear/racturc

calization studies based on the classical kinematic hardening solid have shown a similar behaviour, indicating that kinematic hardening may be used to approximately model a material that develops a rounded vertex on the yield surface (Tvergaard, 1978; Hutchinson and Tvergaard, 1981).

The analysis of shear localization in a homoge- neously stressed solid is straightforward, whereas a complex numerical analysis is required to study the development of localization in a solid subject to a non-uniform state of deformation. The first numerical study of this kind focussed on localization in the neck of a plane strain tensile specimen (Tvergaard et al., 1981) based on ./2 corner theory, and other similar analyses have considered a plate subject to pure bending (Triantafyllidis et al., 1982), a thick-walled tube under internal pressure (Larsson et al., 1982), and the deformation fields near a blunting crack tip (Needleman et al., 1983). In analyses based on the Gurson model, localization has been predicted in the matrix material between two larger voids (Tvergaard, 1982b), in a notched tensile specimen (Needleman et al., 1984a), and at the tip of a growing crack (Needle- man et al., 1987).

In the present paper the combined kinematic/ isotropic hardening model of a progressively cavi- rating material is used to analyse localization and shear fracture at the surface of a bent specimen. Here, the inhomogeneity of the fundamental deformation is clearly less extreme than that at a crack tip; but on the other hand bending is a rather clean problem, suitable to study the basic influence of the nonhomogeneous deformation on the development of localized shearing. Predictions for kinematic hardening are related to those for isotropic hardening, and the results are related to a previous investigation of ductile shear fracture at the free surface of a plane strain tensile specimen.

2. Kinematic hardening porous material

The material model to be used here is a kinematic hardening version of the constitutive relations suggested by Gurson (1977a, b). This model was first suggested by Mear and Hutchinson (1985)

for a porous ductile material, and Tvergaard ( 1987 ) extended the model to account for the nucleation oI' voids.

The model makes use of a family of isotropicj kinematic hardening yield surfaces of the form q)(o':, c(:, %, f ) = 0, where f is the current void volume fraction, o ~: is the average macroscopic Cauchy stress tensor and o~" denotes the center of the yield surface. The radius % of the yield surface for the matrix material is taken to be given by

o r = (1 -- b )Oy -~ bo M {2.1)

where o v and o M are the initial yield stress and the matrix flow stress, respectively, and the parameter b is a constant in the range [0, t]. The expressions are chosen such that for b = 1 they reduce to Gurson's (1977a, b) isotropic hardening model, whereas a pure kinematic hardening model appears for b = 0.

The approximate yield condition is here taken to be of the form

a) [ i q~ = -7 + 2q l f * cosh,[ o~ 2or )"

(1 + (q , f * )2 )=0 (2.2)

where 6 '~ = o 'J - a 'J, ff~ = (3.~USil)I/2 and 'J = ff'/ l~ij~k For f * - ~,., o k . = f and q~ = 1 the expression

(2.2) is that proposed by Meat and Hutchinson (1985), which coincides with that of Gurson (1977a, b) if b = 1. The additional parameter q, was introduced by Tvergaard (1981, 1982a), who found that the agreement with numerical studies of materials containing periodically distributed circular cylindrical or spherical voids is consider- ably improved by using ql = 1.5.

While the value q~ = 1.5 is applied to improve the predictions at small void volume fractions, the function f * ( f ) in (2.2) has been introduced by Tvergaard and Needleman (1984) to model the complete loss of material stress carrying capacity due to void coalescence that occurs at somewhat higher void volume fractions. The function was chosen as

,i/, f, for f~ \

(2,3)

I/i Tvergaard /Duct i leshear f rac ture 33

where the void volume fraction at final fracture is denoted by fv, so that f * ( fF )=f~ = 1/q l (see Fig. 1). According to (2.3) the modification of the yield condition due to the effect of coalescence starts when the void volume fraction reaches a certain critical value fc. Based on experimental results and numerical model analyses the values fc = 0.15 and fv = 0.25 were chosen by Tvergaard and Needleman (1984).

All equations are given in the context of a Lagrangian formulation of the field equations in which a material point is identified by the coordinates x i in the reference configuration. The metric tensors in the current configuration and the reference configuration are denoted Ggj and gij, respectively, with determinants G and g, and ~j denotes the Lagrangian strain tensor. The con- travariant components of the Cauchy stress tensor rr ~s and the Kirchhoff stress tensor ~? on the embedded deformed coordinates are related by the expression T 'j = G~ o ij. Indices range from 1 to 3, and the summation convention is adopted for repeated indices.

The plastic part of the macroscopic strain increment ~/P and the effective plastic strain increment ~P for the matrix material are taken to be related by (see Tvergaard, 1987)

8iJ~Ptd = (1 - f ) OFf ~ . (2.4)

For f= 0 (2.4) is an exact relationship for the classical kinematic hardening solid, and for b = 1

'6e I d F 1.0

0.5

k F

Fig. 1. Yield surface dependence on the function f * in (2.3), the stress components f lu = o,s _ au, and the parameter o F.

the expression reduces to the equivalent plastic work expression applied by Gurson (1977a). Sub- stituting the uniaxial true stress natural strain curve for the matrix 1/E)6M, into (2.4) gives

material, fe M = (1 /E t -

EE t 6i;ilP; OM = E -E t (1 - f )o v (2 .5)

where E and E t a re Young's modulus and the tangent modulus, respectively.

As in the Gurson model the change of the void volume fraction during an increment of deformation is taken to be given by

/---- (/)growth q- (/)nucleation" (2.6)

Since the matrix material is plastically incompressible the increment due to growth is given by

(/)growth = (1 -- f ) GiJiJ~j. (2.7)

Nucleation of new voids occurs mainly at second phase particles, by decohesion of the particle-matrix interface or by particle fracture. As suggested by Needleman and Rice (1978) the increment due to nucleation is taken to be given by

1 k " (/)nucleation = d~OM q- 3~(Ok ) " (2 .8)

A fictitious Gurson yield surface q)o = c/ic(o~, o M, f ) was used by Tvergaard (1987) to formulate the constitutive relations, where o M and f are the current values, and o~ J are a set of fictitious stress components chosen such that

oLJ ~'J - (2 .9 )

O M O F

With this assumption, ~o- 0 is a direct consequence of = 0. In general, the fictitious stresses o~ j differ from the actual stresses o 'J at every point of the current yield surface.

e in a point of the yield The expressions for ~/ij surface ~---0 is chosen identical to that given by the Gurson model in the corresponding point of the fictitious surface ~c = 0. Thus, the plastic part of the macroscopic strain increment is taken to be

-~mi /mkto (2.10)

56 U. Tt,ergaard /" Ductile ~'hear /)'u~ttm

where

l/G -- 3 Si] m - - + o~Gij ,

o F O F

(2.l l)

/ -k ) sinhl k- a= ~f*ql /2v '

~, Orb /~ = OL q- 60,~O" M Of "

(2.12)

H = OM [--3a(1 - f Orb of

o v Orb ~ EE t o" M O(I F f E - E~

~ + a k . (2.13) O F ]

Plastic yielding initiates when rb = 0 and 4)> 0 during elastic deformation, and continued plastic loading requires

1 F Vkl rb=0 and ~mk:o .>t0.

The evolution equation for the yield surface centre during a plastic increment is taken to be

,~" = ~6's, ~>.>0 (2.14)

which is a finite strain generalization of Ziegler's (1959) hardening rule (see Tvergaard, 1978). The value of the parameter/ i is determined so that the consistency condition, q) = 0, is satisfied

~=(1-b +~- - O F

2 Oy O~ ~ Vkl rn~Ssk/+ Ov Of 3Gk:

Oy 1 Orb{ + 3,~(1 - f )

Ov H 3f

1_:1 m:

(2.15)

In cases where large rotations of the principal stress axes occur relative to the material, the formulation of (2.14) in terms of the Jaumann rate may give a poor representation of material be haviour. Then. other corotational rates may be preferable, as has been discussed by Dafaiias (1983) and Lee et al. (1983): but this is no problem in shear localization studies, where the rotations of the principal stress axe~ prior to [ocahza- tion are quite small.

Nucleation controlled by the plastic strain J:~ modelled by taking ,~> 0 and ~d= 0 in (2,8k assuming that nucleation follows a normal distribution as suggested by Chu and Needleman (1980).. Thus, with the mean strain for nucleation

V. Tvergaard / Ductile shear fracture 57

gion is considered here, since the focus is on the development of localization and shear fracture. For a plate subject to pure bending Triantafyllidis et al. (1982) have found that J2 corner theory predicts shear band formation on the compressive side of the plate prior to the tensile side. However, in the case to be studied here, where micro-voids rather than yield surface vertex effects are the dominant mechanism in the plate material, localization occurs late on the compressive side, due to little void nucleation and complete lack of void growth, so that shear bands are first expected on the tensile side. Therefore, the surface region to be considered here is on the tensile side of the bent plate.

The plate surface is taken to have an initial waviness given by

yx l w = - ~80 cos 10 (3.1)

For such a uniformly bent plate, there are solutions symmetric about the planes xl = 0 and x 1 =

x2~

t o

x 1

free surfuce

I 1

I L

{Q) {bl

Fig. 2. Surface region analysed numerically. (a) Initial geometry. (b) Deformed state.

l 0. This type of solutions will be studied here by numerical analyses for the region shown in Fig. 2(a). At the bottom of this region, x 2= -h 0, there is no exact symmetry condition, but here the material is assumed to slide freely on a cylindrical surface, which remains normal to the two sides of the region, at x 1 = 0 and x I = l 0. The radius of the cylindrical surface is denoted R = R(O), and the point (x l, x2)=(0 , -ho) remains fixed. Then, after bending to an angle 0 the deformed region has a shape as that shown in Fig. 2(b). Expressed in terms of the Cartesian reference coordinates x i, the displacement components u i and the nominal traction components T ~ the boundary conditions are

T '=0 fo rx2=w(x l ) , (3.2)

u 1=0and T 2=0 fo rx 1=0, (3.3)

(u I + x 1) cos 0 ]

- ( u2 + x2 + R + ho )sin S = O, ) f rx l=/0 ,

T 1 s in0+T2cos0- -0 ,

(3.4)

(X 1 q- b/l) 2 q- ( l l 2 q- R ) 2 - R 2 = 0 , ]

Tt R + U 2 2xl + Ul t for x 2 = -h 0. R 7" =0,

(3.S)

The present analysis is analogous to that carried out by Tvergaard (1982c) for a surface region in a solid subject to uniaxial plane strain tension. However, in the present analysis the surface region is subjected to a strain gradient, so that the highest strains occur near the surface. This means that the condition for the onset of shear bands is first met in the material near the surface, at a stage where the material remote from the surface is still far from loss of ellipticity.

The angle 0 is taken to be the prescribed quantity, and the variation of the radius R = R(O) of the cylindrical surface at x 2 = -h 0 is specified as a function of 0, which is here taken to be

R = R o + lo/0. (3.6)

As a consequence of (3.6) the average linear strain in the fibre along x2= -h 0 grows proportionally

58 V. Tvergaard / Ductile shear fi'aclure

with 0, while the average logarithmic strain in this fibre is ~ = ln(1 + ORo/lo). With the relationship (3.6) the case of a uniformly strained surface region corresponds to the limit of choosing the constant R 0 very large relative to h 0.

For comparison it is noted that Triantafyllidis et al. (1982) analysed pure bending by considering a full surface-to-surface region of the plate. In the present procedure this would correspond to choosing h 0 as the initial plate thickness and replacing the boundary conditions (3.5) by traction free boundary conditions analogous to (3.2).

The numerical analysis is based on a Lagrangian convected coordinate formulation of the governing equations, as mentioned in Section 2. The Lagrangian strain tensor is given by

~,= ~(u,,, + u,,, + u~u,, , ) (3.7)

where u s are the displacement components on the Cartesian reference base vectors, and ( ),~ denotes covariant differentiation in the reference frame. All cases considered are subject to plane strain conditions, so that u 3 = 0.

The equations of equilibrium are specified m terms of the principle of virtual work, and a linear incremental solution procedure is employed. The equations governing the stress increments +", the strain increments "h~j, etc., are obtained by expan- ding the principle of virtual work about the current state, using (3.7). The incremental equation is to lowest order

~,{ 4ijS~qi; + riJh~Su,,/ dV

(3.8)

where V and S are the volume and surface, respectively, of the region analysed in the reference configuration. The terms bracketed in (3.8) give a small correction if equilibrium is not exactly satisfied in the current state. The boundary condi, tions corresponding to the incremental equation (3.8) are obtained by a similar expansion of (3.2) to (3.5) about the current state. Also here the correction terms analogous to that in (3.8) are

retained to prevent drifting ol the soluuon away from the true conditions on equilibrium and geometry at the boundaries.

The mesh used for the finite element analyses consists of quadrilaterals, each built up of tkmr linear displacement triangular elements. A uniform mesh is used, and most of the computations to be discussed have 8 quadrilaterals in the .v'~-di - rection and 48 in tile x~'-direction. Previous numerical investigations of shear band localization have shown that a careful mesh design is needed in order to get accurate predictions (Tvergaard, Needleman and Lo, 1981: Tvergaard, 1982b). Therefore, the initial aspect ratio of the quadrilaterals to be used here is chosen such that bands forming along the diagonals at the appropriate critical strain will have the critical angle of inclination.

The details of the numerical method have been described elsewhere (Tvergaard, 1982c: Needle- man and Tvergaard, 1984b) and shall not be repeated here. Final fracture is represented by the element vanish technique introduced by Tvergaard (1982c). According to (2.2) and (2.3) the material has lost all stress carrying capacity when the void volume fraction reaches the value fr- In the present computations an element is taken to vanish (the element stiffnesses are neglected) when .f reaches the value 0.9fv and the node forces corresponding to the remaining stresses in this element are stepped down to zero during a number of subsequent increments.

If the material were incompressible, and if 3~, = 0 in (3.1), the logarithmic strain % on the surface, as a function of 0, could be derived directly from incompressibility

%= ~ln -~ +20 . (3.9) (J

This expression is not valid for the material considered here, which shows plastic dilatancy (due to void growth) in addition to elastic dilatancy. How- ever, the strain (3.9) remains a reasonable ap- proximation in most of the range of interest, and it is preferred in the following to plot the development of localized strains as a function of ~0 rather than 0.

I/. Tvergaard / Ductile shear fracture 59

The incremental solutions are obtained for prescribed increments of the angle 0 in (3.4) and (3.6). However, in one case (Fig. 12) it turns out that there is no static equifibrium solution for which the angle 0 increases monotonically. In this case a special finite element Rayleigh-Ritz method is used to follow the non-stable equilibrium solution during the relatively short intervals, in which

is negative while the shear band keeps growing (see also Tvergaard, Needleman and Lo, 1981; Tvergaard and Needleman, 1984). The actual behavior in such intervals would be a dynamic snap-through, but here only quasi-static solutions are considered.

4. Results

In the first case to be analysed the material is taken to have plastic strain controlled nucleation, characterized by the parameters fN = 0.04, c N = 0.3 and s=0.1 in (2.16), and the initial void volume fraction is fl = 0. The uniaxial true stress- logarithmic strain curve for the matrix material is represented by the piecewise power law

directly comparable with those found previously. Bifurcation into diffuse modes as well as loss of ellipticity was also investigated for this material (Tvergaard, 1982c), and it was found that loss of ellipticity and bifurcation into a surface wave mode occur nearly simultaneously, at c I = 0.229 and E t = 0.228, respectively. Thus, in contrast to the analysis based on J2 corner theory (Triantafyllidis et al., 1982), the initial surface waviness (3.1) assumed here can hardly trigger the surface wave instability prior to the onset of shear bands. How- ever, the highest surface strains develop at the wave bottoms, and therefore the shear bands start to grow from these points.

Figure 3 shows the development of the maximum principal logarithmic strain in two material points vs. the parameter % defined by (3.9). Both material points are in the middle of the region analysed, at x 1= l 0, with (A representing the elements just below the surface and E B representing elements at some distance below the surface, through which a shear band grows at a later stage of the process. The strain development

~ o/E , for o ~< Ov, oy( o 1 (4.1)

t~- [~vv ] , fo ro>ov

where E, o v and n are Young's modulus, the initial field stress and the strain hardening expo- nent, respectively. These values are taken to be given by ov/E = 0.0033 and n = 10, Poisson's ratio is v = 0.3, and the additional parameters in (2.2) and (2.3) are ql = 1.5, fc = 0.15 and fv = 0.25.

The amplitude of the initial surface waviness of the form (3.1) is 30 = 0.005/0, and the initial aspect ratio of the region considered is ho/ lo = 10.2. Furthermore, the constant in (3.6) is chosen as R 0 = 0, so that the average strain in the bottom fibre remains equal to zero.

For uniaxial plane strain tension the same material has been analysed previously (Tvergaard, 1982c). Since the same initial aspect ratio of the quadrilaterals is used, the localization predictions in a nonuniform strain field to be found here are

1.0 O.B

0.6

0.t,

0.2

i 1 I /

\ ~ \"~kx, " /~B

j ,

/ ,," ....

: ;';- "~o: s2 o

/ '~ju:l.(.o

02 03* 0.6

~ b : O

- - - - - -b=O.S

........ b=1 i

0.8 g0 1.0

Fig. 3. Max imum pr inc ipal logar i thmic strains at two mater ia l points vs. to, when the initial aspect rat io is ho/ l o = 10.2. The mater ia l has strain contro l led nuc leat ion fN = 0.04, ~N = 0.3, S = 0.1, and no initial voids, f l = 0.

60 L 77wrgaard Duc tdc ~hetxr era+ l~r:'

is shown for three different computations corresponding to b = 1, b = 0.5 and b = 0, respectively In the initial stage, for c 0 < 0.25, the deformation pattern remains close to uniform bending, and here it is seen that the surface strain E a is well approximated by the value %, which was obtained by assuming incompressible material behaviour In this initial stage e R is smaller than CA, as would be expected in uniform bending; but subsequently ~u grows larger than E A, while localization starts to occur, and ~A becomes significantly smaller than e o-

During the initial uniform bending stage plastic yielding takes place in the whole region analysed apart from a thin layer near the bottom (x z -h0) . In Fig. 3 an arrow indicates the point at which the first elastic unloading takes place any- where in the top half of the region analysed. Subsequently, the unloading region spreads, and e~, remains essentially constant after that unloading has reached the middle surface point. Locali- zation occurs along a mesh diagonal, and Fig. 3 shows the angle of inclination ~,~ of this diagonal corresponding to first elastic unloading. The critical angle of inclination corresponding to bifurcation into a shear band is 44.5 for the material considered here subject to uniaxial plane strain tension (Tvergaard, 1982c). It is noted that the value ~ = 44" found for b = 0 is just below the critical value corresponding to bifurcation, whereas the inclinations ~b, found for b = 0.5 and h = 1 are higher.

The earliest localization in Fig. 3 is predicted by the material model with full kinematic hardening (b = 0), and the localization predicted for h - 0.5 is not much later. In both cases shear localization initiates at a surface strain significantly larger than the critical value 0.229 corresponding to uniform plane strain tension, so clearly localization is delayed by the nonuniform deformation field. This delay is particularly strong for b = 1, where ~t,, = 52 is well above the critical angle for shear bands, resulting in a much delayed final fracture, which occurs along the line x ~= 0 rather than inside the shear band.

Figure 4 shows the mesh in the initial state and at three of the subsequent deformed stages for the computation with b = 0.5 in Fig. 3. In Fig. 4(b), at

~,~ IL306, localization has not yet started, whereas the deformed meshes in Figs. 4(c) and ~d) clearly show the development of a shear band near tile free surface. Figure 5 shows curves of constant maximum principal logarithmic strata ~: and void volume fraction f, respectively, at three stages of the deformation. The peak strains shown here inside the band are not quite as high as those indicated m Fig. 3 for the same value of {,. The values of ~u in Fig. 3 are the actual peak strains found in the computations, whereas the contour plots are based on average strains within each quadrilateral.

l'he distribution in Fig. 51a) correspond to the deformed mesh in Fig. 4(b), and here the devia- lions from uniform bending are still rather ~,mait. even though ~0 =-0.306 is well above the critical strain for shear bands under uniform tensile dr:- formations. In Fig. 5(b), at ~ = 0.357. a shear band is well developed, and at % : 0.437, i'ailure by coalescence in a void-sheet has started to occur. It is noted that the deformed meshes and the distri- butions of e and f found for h ..... 0 are very similar to those for h=0.5 shown Jri Figs. 4 and >. whereas the results corresponding to l, = 1 m Fig. 3 are rather different, as discussed above

The computations illustrated in Figs. 3~ 4, and 5 have been repeated with a differeni initial aspect ratio ]G/I~, = 11.5 of the region analysed, but with all other parameters unchanged, Thus, the initial angle of inclination of the diagonals is smaller for this mesh than for the mesh considered in the previous figures, and therefore the diagonals reach the critical angle for shear bands at a larger strain. All three values of ~ found in Fig. 6 are smaller than those found in Fig. 3. For b - 1 this resuh:s in much earlier localization, leading to shear irac- ture inside the shear band (in contrast to the fracture mode found in Fig. 3 for b = 1). For b = ,,! the predicted shear localization is not much af- fected by the different inclination of the diagonals, but is slightly delayed. Simple model analyses have shown that the localization strains predicted by the Gurson model with isotropic hardening are much more sensitive to the inclination of the shear band than predictions based on the material model with kinematic hardening (Mear and Hutchinson. 1985; Fvergaard, 1987). The same trend is seen by


i l l l l i l l i l l I i l i l i i

Hlllllll i ] l l i l l i l I l i l J t l l l I I I I l i l l l

I I I I l l l l l

i i l i l l~ 111111111 i ] l l i t l ] i I l i l i f l ] l ~ l l l l l l l Ill[liB IIIII1111

~11t111 I [ [ ] l i i l i I l l l [ [ l l l l l l l

iilll HlllltH I l l i [ l ] [ i [ l i i l l [ l l

Htlil[li [ l i l l i [U I l l i l l i l /

fH tttH I l l l l l l l i

Co) (b) (c) (d)

Fig. 4. Ca) Initial 8 48 mesh with aspect ratio ho/l o = 10.2 and imperfection ~o/Io = 0.005. (b) Deformed mesh at c o = 0.306, (c) c o = 0.404, (d) c o = 0.437. The material has b = 0.5, strain controlled nucleation fN = 0.04, c N = 0.3, s = 0.1, and no initial voids, f[ = 0.

0.1

0.5

0 .3~f ! /

J / /c=o.3

\0,2

--4!__

0.~

\0.05

=0.05~

~,0.01

0.6

.0.5

, ~ 4 0.3

0 .05)

/ /-

)f=O.O!

j /~ .01

J

-0.1

(Q) (b) (c)

Fig. 5. Curves of constant max imum pr incipal logar i thmic strain E and void volume fract ion f . The initial aspect rat io is ho/ l o = 10.2. (a) % = 0.306. (b) c o = 0.357. (c) c o = 0.437. The mater ia l has b = 0.5, strain contro l led nuc leat ion fN = 0.04, C N = 0.3, s = 0.1, and no initial voids, f l = 0.

62 Pt 7~,ergaard / Ductile vhear lraclurc

1.o U- I

/

i

0.6 / !, / i

i

O J, , eA [ &7

0.2 / - - '~ "xX"-----Wu = 43~ - -b=O

______ b,=O, 5

. . . . . . . . b=l

0 L i 0 0,2 0.4 0,6 0.0 ~0 1.0

Fig. 6. Max imum pr inc ipa l logar i thmic stra ins at two mater ia l

po ints vs. %, when the initial aspect rat io is bo l l o = 11.5. The mater ia l has stra in cont ro l led nuc leat ion ]'N = 0.04, e N = 0.3,

s = 0.1, and no initial voids, f [ = O.

compartson of Figs. 3 and 6, with the significant consequence that predictions of localization m a nonuniformly deformed solid are much less sensitive to the mesh design in the case of kinematic hardening.

The development of the void volume fraction is illustrated in Fig. 7, corresponding to the case b = 0 in Fig. 6. The first two stages shown are prior to localization, while the subsequent three stages illustrate the localized growth of voids inside the growing shear band. The triangular ele~ ments, in which failure has occured, are painted black in the figures, thus illustrating the development of a void-sheet fracture by coalescence of the voids inside the shear band. It is noted that the void volume fraction remains very low ( /< f).!)01) at the bottom of the region analysed, where the strains are relatively small.

Results for a material with a smaller (maybe more realistic) volume fraction of void nucleating

o.fo01

0.01

f= o.oo__~

I L)O'OS

,"1 / I " ,

}0.01

/ /0 O1

f=/ 0.0Ol

I J _ J

o11 '!N ! 0.05.,'" ~

/ (// ' /

/

/o.oll J !

f: i o.oo1, i

i [

i J

(G) (bl () (d) (e)

Figi 7. Curves of constant vo id vo lume f ract ion f . The initial aspect rat io is ho/ l o =11.5 . (a) % =( I .159. (b) % = 0 . 2851 ( C ) % = 0.361. (d) ~0 = 0.381. (e) % = 0.408. The mater ia l has b = 0, stra in cont ro l led nuc leat ion fN = 0.04. ~N = 0.3, s = 0.1, and no

init ial voids, f j = 0.


particles, fN = 0.01, but with all other material parameters unchanged, are shown in Fig. 8. The initial aspect ratio of the region analysed is h o/l o -- 19.5. Here, more void growth is required prior to localization, and for b = 0 the first elastic unloading occurs at a relatively large strain E 0 = 0.59, with ~b u = 42 . For b = 0.5 localization occurs significantly later, with ~b u = 47 o, and for b = 1 the computation is stopped at a so large angle of inclination of the diagonals, +- -54 o, that localization will hardly be predicted based on the present mesh. These results are somewhat analogous to those shown in Fig. 3, but clearly in Fig. 8 the localization strain is much more sensitive to the yield surface curvature at the point of loading. Based on the comparison of Figs. 3 and 6 it is expected that localization would be predicted for b = 1 if the initial aspect ratio of the quadrilaterals were chosen so that the initial angle of inclination of the diagonals is smaller.

1.2 , i i ,

1,o C B ~

o.,

/ /

0 i I I I

0.2 0.4 0.6 0.8 ~0 1.0

Fig. 8. Maximum principal logarithmic strains at two material points vs. %, when the initial aspect ratio is bol l o = 19.5. The material has strain controlled nucleation fN = 0.01, c N = 0.3, s = 0.1, and no initial voids, f l ~ 0.

Figure 9 shows the development of the void volume fraction corresponding to the case b = 0 in Fig. 8, starting at an early stage where very few voids have nucleated and ending at a stage where shear fracture has developed from the free surface. In Fig. 9(e) the continuation of the shear band is not visible, as it is in Fig. 7(e) in terms of the f = 0.05 contour. Due to the smaller initial inclu- sion concentration the localized strain inside the shear band has to grow relatively larger before f = 0.05 is reached.

A material with stress controlled nucleation according to (2.17) has also been investigated. The volume fraction of void nucleating particles, the mean stress for nucleation, and the corresponding standard deviation are taken to be fy = 0.04, a N = 2.1o v, and s = 0.40 v, respectively, so that the material is identical to one analysed previously for conditions of uniform plane strain tension (Tvergaard, 1982c). For this material subject to uniaxial plane strain tension it was found that the critical strain for localization is 0.237 with the corresponding angle of inclination ~b c = 42.8 of the shear band. However, the strong non-normality of the plastic flow law in the presence of stress controlled nucleation results in a complex post-bifurcation behaviour. Thus, shear failure does not occur in the shear band first critical, but at a somewhat larger strain around 0.28 (see Tvergaard, 1982c, Appendix).

Numerical results are shown in Fig. 10 for a region with the initial aspect ratio ho/lo = 12.0. For b=0 and b=0.5 localization leading to void-sheet fracture inside the shear band is predicted, whereas for b = 1 elastic unloading occurs at a too large angle of inclination q~u = 45 o, so that here failure grows along the xZ-axis, as was also found in Fig. 3. The previous numerical analysis for a surface region under uniaxial plane strain tension gave initial unloading at the average strain 0.266. Thus, the initial unloading predicted in Fig. 10, ranging from ~0 = 0.32 for b = 0 to % = 0.42 for b = 1, shows a significant delay in localization due to the nonuniform deformations.

Figure 11 shows void volume fraction distribu- tions corresponding to the case b = 0 in Fig. 10. The much earlier and more widespread nucleation of voids resulting from the stress controlled

64 k Tuer~aard / Ductile shear/kac~urc

0.01

o.oo__!

/ !

f

f= 0.00_ I

__ J

0.05 L_..~ / / i

i /

!

0.01 _/ i - - - J /

i r

L

i f

i j

i' i i

J i

i .i

o.oi ! i

i i

I I

I I

(a) (b) (c) (d) (e)

Fig. 9. Curves of constant vo id vo lume f rac t ion f . The init ia l aspect rat io is ho/ l o =19.5 . (a) t 0 = 0.201. (b) q, = 0.565. (c) % = 0.692. (d) q~ = 0.720. (e) ~0 = 0.790. The mater ia l has h = 0, s t ra in cont ro l led nuc leat ion fy = 0.01, ~ = 0.3.. ' ; = 0.] , and not in it ia l voids, f l = 0.

1.0 , i - -

C

0.8

0.6

0.4

0.2

0

I /

I cB

/ / //'

\~ _41o _ _ _

0 0.2 0.4 0.6

- - b = O

b=0.5

. . . . . . . . b=l

i

0.8 0 1.0

Fig. 10. Max imum pr inc ipa l l ogar i thmic s t ra ins at two mater ia l

po in ts vs. Co, when the init ia l aspect ra t io is ho/ l o = 12.0. The mater ia l has stress cont ro l led nuc leat ion fN = 0.04, O N =

2.1o v , s = 0 .4Or , and no init ia l vo ids , f I = 0.

nucleation criterion is illustrated by Fig. l l(a), where the strain level E 0 = 0.154 is close to that in Fig. 7(a). Even Fig. 5(a), corresponding to a much larger strain level than Fig. l l(a), shows much less porosity. Also in the two subsequent stages, Figs. l l (b) and (c), the width of the band enclosed by the f = 0.05 contour tends to be larger than found in the previous figures.

All the analyses so far have presumed a periodic pattern of shear bands, as implied by the periodicity conditions (3.3) and (3.4). The same assumption was used in the earlier analyses for a surface region of a solid subject to uniaxial plane strain tension (Tvergaard, 1982c). However, if there is a periodic pattern of shear bands across an otherwise uniformly strained specimen, bifurcation will clearly occur into just one shear band (or two crossing shear bands), since a small increment of this alternative mode of deformation requires less external work than that corresponding

ld Tvergaard / Ductile shear fracture 65

iI 0 . 0 0 "~

7

f= 0,01

0.05

I

(o (bl (c)

Fig. 11. Curves of constant vo id vo lume f ract ion f . The initial

aspect rat io is ho/ l o =12.0 . (a) % = 0.154. (b) % = 0.361. (c) o = 0.433. The mater ia l has b = 0, stress contro l led nuc leat ion

fN = 0.04, a N = 2 .1or , s = 0 .4o v , and no initial voids, f l = 0.

to the same increment of elongation in the periodic mode.

For a periodic pattern of shear bands starting from the surface of a bent specimen the question of stability is less clear; but it is expected that bifurcation into a smaller number of active shear bands will occur at some stage of the deformation. In fact this problem is somewhat analogous to the instabilities in the growth pattern of a system of straight edge cracks in a brittle solid, studied by Nemat-Nasser, Sumi and Keer (1980). As long as the spacing of such cracks is large compared with the common crack length, there is very weak interaction between adjacent cracks, and the uniform growth of all cracks is stable. However, the interaction between adjacent cracks becomes more im- portant as the common crack length increases, and at some critical length to spacing ratio some cracks stop growing, while the others grow at a faster rate.

To investigate the stability of the growth of a periodic pattern of shear bands from the free surface, two additional analyses are carried out here. In both cases the length of the region analysed is taken to be 2/0, twice that considered previously, so that now two shear bands grow in the region analysed. This will not give a complete picture of the growth of a periodic pattern of shear bands, but it will allow for the possibilities that either both bands grow simultaneously, or one band stops and the other grows alone. For the surface region with length 21 o and height h 0 the boundary conditions (3.4) are now applied at x 1 = 2l o, and the initial surface waviness (3.1) is re- placed by

,I'rX 1 qTX 1 w = - 8 o cos~-,o 8, cos 2l--~" (4.2)

If 81 = 0, this surface waviness is identical to (3.1). The additional waviness with amplitude 81 is only introduced to give a small imperfection, which will slightly favour growth in one of the shear bands.

In the first computation the initial geometry is identical to that used in Fig. 3 (except for the double length 2/0), and the material is that corresponding to b = 0 in Fig. 3. Thus, the initial aspect ratio is defined by ho/l o = 10.2, and due to the double length a 16 48 mesh is now used, so that the initial aspect ratio of the quadrilaterals is identical to that used previously. The amplitudes defining the surface waviness are chosen as 60 = 0.005/0 and 81 = 0.0005/0. Thus, there are two wave bottoms, at x ~ = 0 and x 1 = 210, from which shear bands will tend to emanate. Figure 12 il- lustrates the results of this computation at two stages corresponding to % = 0.313 and % = 0.332, respectively.

The stage in Fig. 12(a) is slightly after the initial elastic unloading, where localization starts according to Fig. 3. Here, localized plastic flow takes place inside both of the crossing shear bands, as is most clearly illustrated by the curves of constant maximum principal logarithmic strain. However, already in Fig. 12(a) one band has grown significantly more than the other, indicating a rather strong sensitivity to the initial imperfection represented by the non-zero value of 81. The

66 I /. Tvergaard / Ductile shear jracture

E

I - \~ :2 . --- J ) "


weakest shear band stops growing (by elastic unloading) at c o = 0.316, and subsequently the other band grows into a surface shear crack, as illustrated by the deformed mesh and the contours of constant void volume fraction in Fig. 12(b).

As mentioned in Section 3 the static equilibrium solutions are unstable in small intervals of the incremental solution between the stages shown in Figs. 12(a) and (b). In reality, dynamic snap through would happen in such short intervals, if a monotonically increasing 0 were prescribed. The solutions here are obtained by using a special finite element Rayleigh-Ritz method to prescribe selected node displacements rather than 0, so that the shear band keeps growing, while t~ may be negative in short intervals.

Another computation has been carried out with the surface waviness given by the amplitudes 60 = - 0.005/0 and 8~ = 0.0005/0. Here two shear bands emanate from one central wave bottom at x 1 = l 0. Figure 13 shows the resulting contours of constant void volume fraction at three different stages of the deformation, corresponding to % = 0.323, c o = 0.340 and c o = 0.354, respectively. At the first stage the void distribution appears to be nearly exactly symmetric about the mid plane x ~= l 0. Unloading in one of the two shear bands happens relatively late, at c0=0.337, and subsequently shear fracture develops in the other band, as shown in Figs. 13(b) and (c).

It is clear from the solutions illustrated in Figs. 12 and 13 that the uniform growth of a periodic pattern of shear bands from the free surface of a bent specimen will not remain stable. It is noted that the growth of one of the bands stops at a smaller value of c o in Fig. 12, where the two bands cross one another, than in Fig. 13, where the bands point in two different directions, starting from the same surface point.

5. D iscuss ion

The present numerical studies of shear localization in a nonuniformly strained specimen show a significant delay of localization. The analyses represent the deformation of a surface region in a bent plate, such that the peak strains occur at the

free surface, and below the surface the strain level decays smoothly towards zero. In a uniformly strained solid, localization is predicted as soon as the strain reaches the critical value for loss of ellipticity, but in the present analyses the average strain on the free surface reaches the critical value long before there is any sign of localization. In fact, the critical strain is reached below the surface at a depth well above ten times the shear band width before a localized shear band starts to develop.

All the analyses show significantly less delay for kinematic hardening than for isotropic hardening. Under uniformly strained conditions these two models of a cavitating solid give the same critical strain for localization, but the previous simple model analyses by Mear and Hutchinson (1985) and Tvergaard (1987) have shown that the kinematic hardening solid is much more imperfection sensitive. The present numerical predictions agree with this observation. Also, both the simple model predictions and the present numerical results show that the solid with kinematic hardening is less sensitive to the angle of inclination of the band.

Numerical localization analyses based on J2 corner theory have shown that localization starts soon after the first loss of ellipticity, even in a nonuniform strain field (Tvergaard et al., 1981; Larsson et al., 1981; Triantafyllidis et al., 1982). The porous material model with kinematic hardening can be considered as an approximate model of a material that develops a rounded vertex on the yield surface, with a local curvature equal to that of the initial yield surface (Mear and Hutchinson, 1985; Tvergaard, 1987). Therefore, the result that kinematic hardening gives a smaller delay than isotropic hardening, but still a larger delay than that predicted for a material that develops a sharp vertex on the yield surface, is as expected.

The nonuniformity of the strain fields in the present analyses is not nearly as drastic as that at the tip of a crack (Needleman et al., 1987). During crack tip blunting, localization of plastic flow is part of the failure mechanism in the near tip field, and therefore it must be expected that using the kinematic hardening material model would result

68 V. Tvergaard / Ductile shear/ra~ turc

in the prediction of more rapid crack growth than that found based on the Gurson model with isotropic hardening.

The length-scales in the problem have an im- portant influence on the results obtained. The bands will choose the smallest possible width, which is equal to the element width (see also Tvergaard, 1982c), and therefore the predictions are clearly mesh dependent. Using twice as many quadrilaterals both in the xLdirection and the x2-direction would give half the band width and approximately the same band inclination. Then, reaching the critical strain for localization along a length of the band corresponding to say 10 times the band width would occur earlier, and therefore earlier localization would be expected.

This type of band width dependence of the solutions is illustrated by a computation that has been carried out for ho/l o = 5.1 and R 0 =h0. using a 16 48 mesh and the material parameters corresponding to b = 0 in Fig. 3. This computation gives a reasonably good representation of the behavior in the top half of the region analysed in Fig. 3, but now with half the minimum band width. The first elastic unloading, leading to localization, occurs at c 0 = 0.262, much earlier than the value c 0 = 0.296 found for b = 0 in Fig. 3, but still significantly after that the critical strain 0.229 has been reached at the surface. Clearly the minimum possible band width, relative to the degree of nonuniformity of the strain field, has a strong influence on the localization delay.

The mesh dependence of the localization predictions reflects the fact that the continuum formulation does not provide a natural length-scale to set a minimum band width. For real materials there are such natural length-scales, e.g. the grain-size or the size of small scale voids of inclusions, which will enforce a finite band width. Thus, the most realistic delay of localization due to nonuniform deformations is that obtained by an element size, which represents the actual material length scale.

The wavelength of the surface waviness, relative to the plate thickness, gives a measure of the degree of nonuniformity of the fundamental strain field. The computations illustrated in Figs. 3 to 13 are based on choosing R 0 = 0; but if a positive

value of R 0 were used instead, without changing the aspect ratio ho/lo, this would represent an increased plate thickness for fixed surface wavelength and thus a relatively less nonuniforln strain field. It is expected that this would reduce the delay of localization, and in fact for R c, , :c the surface region would be subjected to uniform straining where there is no delay at all (see Tvergaard, 1982c).

The last tow analyses. Figs. 12 and i3, taking into account the simultaneous growth of two adjacent shear bands, have made it possible to study the stability of a periodic pattern of shear bands growing from the free surface. Both computations show that simultaneous growth of the two bands is stable initially. However, at a later stage, when the bands are so long relative to their spacing that they interact strongly, one of the bands stops growing, while the other band grows faster. In the present computations the bands are not com- pletely identical initially, due to an assumed im-. perfection; but if they were identical, the change of growth pattern would represent a bifurcation from the periodic pattern. This behaviour is analogous to that found for a system of straight edge cracks in a brittle solid (Nemat-Nasser et aJ~, 1980).

Although the studies illustrated in Figs. 12 and 13 give some understanding of the interaction between adjacent growing shear bands, this is not the full picture, Figure 12 shows that the growth of two bands crossing one another gets unstable at a rather early stage, whereas two bands growing in different directions from the same surface point (Fig. 13) can extend further before one of them stops growing. This does indicate a sequence of different instabilities in the growth pattern of a full periodic array of bands. However, a complete understanding of the behaviour would require a much larger computation than that in Figs. 12 and 13. This larger computation would have to account for the full nonlinear interaction of several adjacent shear bands which start off growing at identical rates. One result of the larger computation could be a prediction of the characteristic spacing between the shear cracks that finally develop on the surface of a bent plate when coalescence leads to void-sheet fracture inside the bands that are

V. Tvergaard/Ducti~shearfracture 69

stil l act ive (e.g. see Fig. 10 in Hutch inson and

Tvergaard , 1980).

Acknowledgement

The suppor t of the Dan ish Techn ica l Research

Counc i l th rough grant 16-4006.M is gratefu l ly

acknowledged.

References

Becker, R. and A. Needleman (1985), "Effect of yield surface curvature on necking and failure in porous plastic solids", Division of Engineering, Brown University.

Chu, C.C. and A. Needleman (1980), "Void nucleation effects in biaxially stretched sheets", J. Eng. Mater. Techn. 102, 249.

Dafalias, Y.F. (1983), "Corotational rates for kinematic hardening at large plastic deformations", J. Appl. Mech. 50, 561.

Goods, S.H. and L.M. Brown (1979), "The nucleation of cavities by plastic deformation", Acta Metallurgica 27, 1.

Gurson, A.L. (1977a), "Continuum theory of ductile rupture by void nucleation and growth--Part I. Yield criteria and flow rules for porous ductile media", J. Eng. Mater. Tech- nol. 99, 2.

Gurson, A.L. (1977b), 'Porous rigid-plastic materials containing rigid inclusions--yield function, plastic potential, and void nucleation", in: D.M.R. Taplin, ed., Proc. lnternat. Confer. Fracture, 2A, Pergamon Press, Oxford/New York, p. 357.

Hutchinson, J.W. and V. Tvergaard (1980), "Surface instabilities on statically strained plastic solids", Int. J. Mech. Sci. 22, 339.

Hutchinson, J.W. and V. Tvergaard (1981), "Shear band formation in plane strain", Int. J. Solids Structures 17, 451.

Larsson, M., A. Needleman, V. Tvergaard and B. Stor&kers (1982), "Instability and failure of internally pressurized ductile metal cylinders", J. Mech. Phys. Solids 30, 121.

Lee, E.H., R.U Mallett and T.B. Wertheimer (1983), "Stress analysis for kinematic hardening in finite-deformation plasticity", J. Appl. Mech. 50, 554.

Mear, M.E. and J.W. Hutchinson (1985), "Influence of yield surface curvature on flow localization in dilatant plasticity", Mechanics of Materials 4, 395.

Needleman, A. and J.R. Rice (1978), "Limits to ductility set by plastic flow localization", in: D.P. Koistinen et al., eds., Mechanics of Sheet Metal Forming, Plenum Publishing, New York, p. 237.

Needleman, A. and V. Tvergaard (1983), "Crack-tip stress and deformation fields in a solid with a vertex on its yield

surface", in: C.F. Shih and J.P. Gudas, eds., Elastic-Plastic Fracture: Second Symposium, Vol. 1 -Inelastic Crack Analy- sis, ASTM STP, 803.

Needleman, A. and V. Tvergaard (1984a), "An analysis of ductile rupture in notched bars", J. Mech. Phys. Solids 32, 461.

Needleman, A. and V. Tvergaard (1984b), "Finite element analysis of localization in plasticity", in: J.T. Oden and G.F. Carey, eds., Finite Elements--Special Problems in Solid Mechanics, Vol. 5 Prentice-Hall, Englewood Cliffs, N J, p. 94.

Needleman, A. and V. Tvergaard (1987), "An analysis of ductile rupture modes at a crack tip", J. Mech. Phys. Solids 35, 151.

Nemat-Nasser, S., Y. Sumi and L.M. Keer (1980), "Unstable growth of tension cracks in brittle sofids: Stable and unstable bifurcations, snap-through, and imperfection sensitivity", Int. J. Solids Structures 16, 1017.

Puttick, D.E. (1959), "Ductile fracture in metals", Phil. Mag. 4, 964.

Saje, M., J. Pan and A. Needleman (1982), "Void nucleation effects on shear localization in porous plastic solids", Int. J. Fracture 19, 163.

Shima, S. and M. Oyane (1976), "Plasticity theory for porous metals", Int. J. Mech. Sci. 18, 285.

Triantafyllidis, N., A. Needleman and V. Tvergaard (1982), "On the development of shear bands in pure bending", Int. J. Solids Structures 18, 121.

Tvergaard, V. (1978), "Effect of kinematic hardening on localized necking in biaxially stretched sheets", Int. J. Mech. Sci. 20, 651.

Tvergaard, V. (1981), "Influence of voids on shear band instabilities under plane strain conditions", Int. J. Fracture 17, 389.

Tvergaard, V. (1982a), "On localization in ductile materials containing spherical voids", Int. J. Fracture 18, 237.

Tvergaard, V. (1982b), "Ductile fracture by cavity nucleation between larger voids", J. Mech. Phys. Solids 30, 265.

Tvergaard, V. (1982c), "Influence of void nucleation on ductile shear fracture at a free surface", J. Mech. Phys. Solids 30, 399.

Tvergaard, V. (1987), "Effect of yield surface curvature and void nucleation on plastic flow localization", J. Mech. Phys. Solids 35, 43.

Tvergaard, V. and A. Needleman (1984), "Analysis of the cupcone fracture in a round tensile bar", Acta Metallurgica 32, 157.

Tvergaard, V., A. Needleman and K.K. Lo (1981), "Flow localization in the plane strain tensile test", J. Mech. Phys. Solids 29, 115.

Yamamoto, H. (1978), "Conditions for shear localization in the ductile fracture of void-containing materials", Int. J. Fracture 14, 347.

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Tvergaard MM 1987

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