Computational modeling of material failure

Computational modeling of material failure A Needleman Division of Engineering. Brown University, Providence Rl 02912 Analyses of fracture are discussed where the initial-boundary value problem formulation allows for the possibility of a complete loss of stress carrying capacity, with the associated creation of new free surface. No additional failure criterion is employed so that fracture arises as a natural outcome of the deformation process. Two types of analyses are reviewed. In one case, the material's constitutive description incorporates a model of the failure mechanism; the nucleation, growth and coalescence of microvoids for ductile fracture in structural metals. In some analyses this is augmented with a simple characterization of failure by cleavage to analyze ductile-brittle transitions. The other class of problems involves specifying separation relations for one or more cohesive surfaces present in the continuum. The emphasis is on reviewing recent work on dynamic failure phenomena and the discussion centers around issues of length scales, size effects and the convergence of numerical solutions.

polycrystalline metal, and the use of a continuum approach is justified. The main ingredients required for the theoretical framework are constitutive descriptions of inelastic flow, equations for damage evolution and crack mechanics. The direct prediction of failure has been pursued for diverse materials and from a variety of perspectives. One approach, often termed continuum damage mechanics, see e.g., Kachanov (1958), Lemaitre (1986), introduces one or more phenomenological parameters to characterize failure. The evolution equations and parameters for damage evolution are chosen from experiment and are usually not directly related to the michromechanical processes of failure. Another approach is based on incorporating a model of the failure process into the constitutive description of the continuum in order to make a more direct connection to the physical mechanisms of fracture. Such theories have been developed for microcracking of ceramics, Charalambides and McMeeking (1987) and Ortiz and Giannakopoulos (1990), ductile fracture of metals by void nucleation, growth and coalescence, Tvergaard (1990a), and for failure modes of metal-matrix composites, Needleman et al. (1993). Initial-boundary value problems are formulated and solved where the loss of load carrying capacity emerges as an outcome of computing the deformation history. Analyses carried out within such a framework have given both qualitative and quantitative descriptions of fracture processes.

INTRODUCTION The main focus of quantitative theories of mechanical behavior is the determination of the deformations and stresses in a solid subject to a given history of loading. Failure due to loss of stability, as in buckling under compressive loading or localized necking under tensile loading, can emerge as a direct outcome of a stress and deformation analysis. Fracture, which involves the creation of new free surface, is generally treated by carrying out a conventional stress analysis with a fracture criterion specified separately. There has been increasing interest in developing theories where fracture emerges as a natural outcome of the deformation history. One motivation stems from the progressive nature of fracture in many engineering materials; local failure causes a redistribution of the stress and deformation fields, which affects the course of subsequent failure, which causes a further stress and deformation redistribution, etc. Accounting for this progression is often essential for predicting final failure. Another motivation is to relate phenomenological measures of ductility and fracture toughness to measurable (and controllable) features of a material's microstructure, to provide a basis for the design of stronger and tougher materials. Ultimately, of course, separation takes place on an atomic scale. However, in many circumstances the processes that are key for determining the mechanical performance of engineering materials take place on a much larger length scale, say of the order of a micron or larger, for example, on the size scale of grains in a part of "Mechanics USA 1994" edited by AS Kobayashi Appl Mech Rev vol 47, no 6, part 2, June 1994

Attention in this review is mainly focused on analyses of ductile failure processes and ductile-brittle transitions in structural metals. The fracture mechanisms

S34

ASME Reprint No AMR146 © 1994 American Society of Mechanical Engineers

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Appl Mech Rev vol 47, no 6, part 2, June 1994

involved are plastic void growth and cleavage cracking. Obtaining solutions to these problems involves the integration of complex, path dependent constitutive relations, the resolution of very high gradients, for example as occur in the vicinity of crack tips, the identification of instabilities and capturing the associated emergence of complex patterns of localized plastic flow, and the creation of new free surface. The theme of the discussion here centers around length scales and size effects. Shear band localizations, which often precede ductile failure, are only discussed in passing. A recent review of localization processes in ductile metals is given in Needleman and Tvergaard (1992). Also, details of the numerical solution procedures are not discussed. These can be found in the original papers and in references cited in them. As Rice (1976) has noted predictions of ductility can, at least in principle, be based on a size-scale independent formulation. On the other hand, predictions of toughness, i.e. the resistance to crack initiation and growth, require a characteristic length associated with the material, if only from dimensional considerations. Additionally a material length scale is needed when, as is often the case, shear localization plays a role in limiting ductility. A length scale can enter the initialboundary value problem formulation in a variety of ways; for example, explicitly by modeling features of the material microstructure, such as the size and spacing of the second phase particles that are almost always present in structural metals, or implicitly as is the case in certain dynamics problems. Another approach involves incorporating a material length scale into the constitutive description of the continuum; by using a non-local stress-strain relation or by introducing surfaces of separation characterized by a tractiondisplacement relation. In grid based numerical methods, finite elements or finite differences, a length scale is introduced by the discretization. When numerical analyses of localization or toughness are carried out without some length scale in the initial-boundary value problem formulation, the length scale introduced by the discretization can determine key features of the computed behavior. Hence, in the discussion here attention is given to convergence issues.

Needleman: Computational modeling of material failure

plastic solids, the basis of which is a flow potential, that characterizes the porosity in terms of a single scalar internal variable, / , the void volume fraction

$ = |i + 2gi r ^h ( ^ ) -1 - giv*2 = o (i) where ae and ah are the effective and mean normal stress (computed from the Cauchy stress cr) and a is the matrix flow strength. With /* = 0, (1) reduces to the Mises potential. The parameters q\ and q^ were introduced by Tvergaard (1981, 1982) to bring predictions of the model into closer agreement with full numerical analyses of a periodic array of voids. The bilinear function / * ( / ) was added by Tvergaard and Needleman (1984) to account for the effects of rapid void coalescence at failure,

[f'+Jf^jrif

Ductile fracture in structural metals involves the growth of neighboring voids to coalescence. The voids mainly nucleate at second phase particles, by decohesion of the particle-matrix interface or by particle fracture and then grow through large plastic deformations of the surrounding matrix material. Typically, for structural metals, the size of the void nucleating particles ranges from 0.1 fim to 100 /J,m, with volume fractions of a few percent. Gurson (1975) introduced a constitutive framework for progressively cavitating porous

- fc)

f>fc

The constant /* is the value of /* at zero stress in (1), i.e. /* = I/91. As / — • / / , / * — » /* and the material loses all stress carrying capacity. The inelastic part of the rate of deformation, d p , is given by

In general the evolution of the void volume fraction results from the growth of existing voids and the nucleation of new voids, ucleation

(4)

The rate of increase of void volume fraction due to the growth of existing voids is determined from the condition that the matrix material is plastically incompressible so that fgrowth = (1 - / ) d P : I (5) The contribution resulting from the nucleation of new voids is taken to be either strain controlled, fnucleation

MATERIAL MODEL FOR DUCTILE FAILURE

S35

— Vl,

or s t r e s s c o n t r o l l e d ,

/nucleation

=

B(cre + &h). The expressions for V and B are taken to follow a normal distribution, as suggested by Chu and Needleman (1980). A viscoplastic version of the Gurson (1975) solid was presented in Pan et al. (1983) and thermal softening due to adiabatic heating was introduced in Needleman and Tvergaard (1991a). In any case, due to the presence of micro-voids, the plastic flow is dilational, and pressure sensitive. Furthermore, although the matrix material continues to harden, the aggregate can soften, with the stress carrying capacity of the aggregate eventually vanishing. Background on the Gurson model and

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

S36

Appl Mech Rev vol 47, no 6, part 2, June 1994

MECHANICS USA 1994

on the micromechanical modeling of plastic void growth are given in Tvergaard (1990a) and Needleman et al. (1992). It is worth separating the general features of the framework introduced by Gurson (1975) from the specifics of the particular flow rule. T h e general features are (i) the one parameter characterization of porosity; (ii) the use of m a t r i x incompressibility to obtain the evolution equation for the void volume fraction and (iii) the use of the equivalence of aggregate and matrix rate of plastic work to relate aggregate hardening to m a t r i x plastic response. T h e particulars are the specific form of the flow potential (1) and the characterization of the m a t r i x material. Recent enhancements to the modeling, within this general framework, include the work of Zavaliangos and Anand (1993) on thermoviscoplastic porous solids, Leblond et al. (1993) on a non-local extension of the formulation and Tong and Ravichandran (1993) on inertial effects. Cleavage is modeled by a critical stress criterion. In some early calculations, Tvergaard and Needleman (1986, 1988), this critical stress was taken to be constant throughout the material. In Tvergaard and Needleman (1993), a cleavage model explicitly governed by a critical stress over a critical distance criterion, Ritchie et al. (1973), was employed. T h e material is partitioned into cleavage grains and it is assumed that cleavage failure in a grain occurs when the volume average of the m a x i m u m principal stress over the grain reaches a t e m p e r a t u r e independent critical value. The size of the cleavage grain serves as the material characteristic length. Tensile bars A typical feature of ductile fracture in structural metals is the contrast between the shear fracture mode observed in plane strain tensile specimens and the cupcone mode observed in axisymmetric tensile specimens. Quasi-static calculations carried out using the Gurson (1975) constitutive framework have exhibited this fracture mode transition, Tvergaard and Needleman (1984), Becker and Needleman (1986), Devaux et al. (1992). In another study, Needleman and Tvergaard (1984) carried out calculations for plane strain and axisymmetric notched specimens t h a t were used in the experiments of Hancock and MacKenzie (1976) and Hancock and Brown (1983). T h e calculations indicate that as long as deviations from a proportional history are not too great, the onset of failure is approximately represented by a single failure locus of stress triaxiality versus plastic strain. T h e numerical results also revealed t h a t the role of shear localization in failure of plane strain blunt notched specimens gives a strong enough deviation from proportionality to deviate from such a single curve. A detailed quantitative comparison between predictions and experiment is more complex because of the p a t h dependent and progressive nature of ductile frac-

ture. Becker et al. (1988) compared quantitative predictions of void growth, strength and ductility with detailed measurements in round notched bars. Various notch geometries were studied in order to obtain different stress histories. T h e tensile specimens were machined from partially consolidated and sintered iron powder compacts in order to minimize nucleation effects and focus on void growth and failure issues. The results indicate t h a t this constitutive framework gives reasonably accurate predictions of void growth and of the effects on strength and ductility for a wide range of initial void volume fractions and levels of stress triaxiality. Comparisons of porosity predictions and experimental observations have recently been carried out by Zavaliangos and Anand (1993). T h e effects of progressive microrupture and thermal softening on dynamic ductile failure in notched bars were analyzed in Tvergaard and Needleman (1990). In plane strain, failure involved localization into a shear band. T h e r m a l softening alone was found to give nearly as rapid load decay as the combined effect of thermal and void softening in the early stages of localization, but thermal softening alone did not account for the subsequent complete loss of load carrying capacity. For axisymmetric conditions, as in plane strain, thermal softening by itself was found to give an overall softening. However, the failure mode is quite different in axisymmetric specimens; failure initiation precedes shear band development and occurs at the center of the specimen. 1.25 -r-r-r 1.00

0.75 10 20 40 80

0.50

Elements Elements Elements Elements

0.25

0.00 0.00

0.02

0.04

0.06

0.08

0.10

U/L Figure 1. Dynamic stress-strain curves obtained using various mesh resolutions in a one dimensional model problem. From Needleman (1992). A systematic mesh convergence study was carried in Needleman (1992) for a simple one dimensional model problem. T h e material was taken to be viscoplastic and strain softening, with the strain softening characteristics mimicking those of a porous plastic solid. Figure 1 shows overall stress-strain curves for various mesh resolutions. T h e curves obtained with 40 elements and with 80 elements virtually coincide. W i t h only 10 elements

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Appl Mech Rev vol 47, no 6, part 2, June 1994

the predicted onset of failure occurs at U/L = 0.081 as compared with U/L = 0.075 with 40 or 80 elements. An important feature for convergence is that the loss of stress carrying capacity takes place over a narrow strain range, as is modeled for a porous plastic solid by (2). As a consequence of this and of the fact that time necessarily increases monotonically (so that snapback is precluded), there is convergence for the value of the time at which the stress carrying capacity vanishes. Dynamic localization gives rise to a continually narrowing boundary layer and, as long as the mesh can resolve this boundary layer, the point at which the sharp stress drop begins will be determined accurately. Size effects Consider a round bar subject to uniaxial tensile loading. Under quasi-static conditions, the response at a fixed strain rate is independent of the size of the bar. The situation is quite different under dynamic loading conditions. Then, the stress carried by the loading wave increases with the imposed velocity, which must be increased with the size of the bar in order to maintain the given strain rate. Hence, different sized specimens subject to the same strain rate experience different stress histories. Additionally, even when wave effects have been damped out, material inertia slows down neck development and lowers the stress triaxiality in the neck center, Needleman (1991). Thus, under dynamic loading conditions the effects of enhanced deformation inhomogeneity, which promotes localized straining in a neck, and the retarding effect of material inertia on neck development compete.

Figure 2. Apparent ductility, as measured by the area reduction at failure initiation, as a function of specimen size. From Knoche and Needleman (1993). Quite general considerations lead to the conclusion that, at fixed strain rate, the dependence of apparent ductility on specimen size is not monotonic. For sufficiently small specimens, the behavior is much like what would be predicted by a quasi-static analysis and

Needleman: Computational modeling of material failure i

'

•

•

*

.^ .

i

Y™J

. ^ .

. *—.

S37 ,

T —r

f 0.10

-

IT/L C -Q.0012

—

0.09

0.04

/

/ /

'''

/ "

/

mash 8*32 -mash 12*48 or 12*96 O.OS

-A'

/ ''

, //~

/

//

LyLc-0.0086

-

0.O2

_ — i — ^ _ , — _ j .

0-60 , , . , . , 0.75

Figure 3. Void volume fraction at the neck center versus the logarithm of the area reduction at that point for various finite element discretizations. From Knoche and Needleman (1993).

is essentially size-independent. As the specimen size is increased, necking and failure are delayed as a consequence of the inertial resistance to motion. Eventually, however, the large stresses associated with the loading wave dominate the failure process and a specimen size will be reached where failure occurs during the initial loading wave and the apparent ductility will vanish. Knoche and Needleman (1993) used the modified Gurson (1975) constitutive framework to analyze this process. A critical void volume fraction criteria was used to identify the initiation of failure. For sufficiently small specimens, the predictions of a dynamic analysis are nearly independent of specimen size and differ little from what would be predicted by a corresponding quasi-static analysis. Then, as the specimen size is increased, the retarding effect of material inertia leads to larger failure strains. However, for a sufficiently large specimen, the failure site shifts from the specimen center to the impact end. During this shift the failure strain decreases, increases and then decreases again. The dashed curve shows the effect of a change in the parameter characterizing failure. Figure 3 shows results using various meshes for two calculations; for the calculation with Lo/Lc — 0.0012 neck development at the mid-plane dominates and there is very good agreement between computations based on the various meshes. However, with LQ/LC = 0.0086 the course of neck development is more complex. In the early stages of necking, the minimum crosssectional area occurs away from the mid-plane where the graduated 8 x 32 mesh is rather coarse. Subsequently, necking and eventually failure initiation occur at the mid-section. The coarse mesh has the effect of promoting deformation at the mid-plane, which is where failure eventually occurs. Thus, the inadequate spatial resolution leads to the prediction of failure at too small a strain. This illustrates that in com-

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

S38

MECHANICS USA 1994

Appl Mech Rev vol 47, no 6, part 2, June 1994

plex nonlinear problems, finite element solutions are not necessarily upper bounds. For sufficiently fine uniform meshes, convergent solutions are obtained.

0.01

0.002

f

C r a c k growth As mentioned in the Introduction, macroscopic measures of resistance to crack initiation and growth, e.g. the critical stress intensity factor, must depend on a characteristic length associated with the material, Rice (1976). For example, the characteristic length may be particle spacing or grain size. In order to avoid an inherent mesh dependence of numerical results, the boundary value problem must incorporate a characteristic length scale. There have been a t t e m p t s to use the discretization length scale as the governing length scale in predicting crack growth. There are inherent difficulties with such an approach. Having the modeling tied to a specific discretization means t h a t notions of convergence, of an underlying continuum solution and of transferability (say between plane strain and three dimensional configurations) are lost. In this section, some recent studies of dynamic crack growth are discussed where a length scale enters the formulation by explicitly modeling the relevant features of the microstructure of the material. As a consequence, the speed of crack growth is directly determined by the micromechanisms of failure incorporated in the material model. This contrasts with the usual approach to the analysis of dynamic crack growth where a critical value of the cracktip-opening displacement, t h e dynamic stress-intensity factor, or some other parameter characterizing the near tip fields is assumed; or where the analyses are based on assuming a constant speed of crack growth. T h e ductile crack growth mechanism analyzed involved two populations of void nucleating particles; larger particles (c=s 10 /jm) t h a t nucleate voids at relatively small strains a n d smaller particles (?« 0.1 yum) t h a t nucleate voids a t much larger strains. T h e small scale particles were taken to be uniformly distributed and nucleate by a plastic strain controlled mechanism. T h e large particles are modeled as "islands" of the amplitude of the void nucleation function corresponding to stress controlled nucleation. T h e r m a l softening due to adiabatic heating was accounted for. T h e initial spacing between the large particles serves as a characteristic length scale. In Needleman a n d Tvergaard (1991ab) attention was confined to large particles lying along the initial crack line. T h e porosity contours in Fig. 4 show the crack growth mode for two initial distributions of the large particles; a uniform distribution, Fig. 4a, and a nonuniform distribution, Fig. 4b. Note t h a t the void volume fraction is negligible away from the fracture surface. W i t h a uniform distribution, the crack opening angle is nearly constant after the initial stages of growth and, in Fig. 4a, is « 7 deg. Figure 5 shows the average crack speed as a function of the root mean square distance between particles,

(a) f

0.01

P'002

^ 7 ^ = % — ^ _

— « • —

«.

(b) Figure 4. Contours of constant void volume fraction for two initial distributions of the larger particles, showing the mode of dynamic crack propagation. From Needlem a n and Tvergaard (1991ab).

.4£r

)

m B

W.AU

0.18

(mm)

Figure 5. Average crack speed versus root mean large particle spacing, DRMSLarger values of DRMS correspond to more non-uniform distributions. T h e dashed line is a linear least squares fit. From Needleman and Tvergaard (1991b).

DRMS]

larger values of DRMS

correspond to greater

deviations from uniformity. T h e dashed line is a least squares linear fit. It is expected t h a t the crack growth behavior would differ significantly for large enough DRMSOne possibility is t h a t crack growth would occur off the initial crack plane so t h a t a measure of the inclusion spacing involving off-crack-plane inclusions becomes t h e relevant length scale. Another possibility, if the off-crack-plane inclusion spacing is large enough, is t h a t the large particles no longer play a key role in the fracture process. As a consequence, some other material length scale would be the appropriate one. Experiments on metal sheets with controlled distributions of holes, Magnusen et al. (1988), have shown t h a t t h e fracture strain for a non-uniform distribution is significantly smaller t h a n t h a t for the same volume fraction of uniformly distributed holes. Also, Becker

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Appl M e o h R ev

vo1 4 7

'

n0 6

' P art 2 '

June 1

"

Needleman: Computational modeling of material failure

4

(1987) has predicted a significant reduction in ductility due to non-uniformities. Further investigations of the effects of porosity distribution on porous plastic response have been carried out more recently, e.g. Benson (1993)- Thus, it may seem contradictory that a nonuniform inclusion distribution increases the resistance to crack growth. However, at a crack tip, non-uniform inclusion spacing may result in early crack propagation through one or two relatively closely spaced voids, but once the next void near the crack tip has a greater than average spacing it is expected that further crack propagation will require a load higher than that corresponding to the average spacing, leading to an increased crack growth resistance for a non-uniform large particle distribution. The effect of specimen size was explored in Needleman and Tvergaard (1991a) and crack growth off the initial crack plane was analyzed in Tvergaard and Needleman (1992, 1993). For a very small specimen that undergoes general yielding, Needleman and Tvergaard (1991a) found that the failure mode is quite different from that for the larger specimens. In Needleman and Tvergaard (1994), a crack growth problem was solved with different levels of mesh refinement near the crack tip to determine whether or not the solutions converged. Crack growth predictions in cases where the large scale voids dominate showed practically no mesh sensitivity, whereas cases dominated by the small scale voids showed a clear mesh sensitivity. For initially sharp cracks the initiation of crack growth was quite sensitive to the mesh. However, for initially blunt cracks the mesh sensitivity of the initiation time is removed. Mesh independent results for creep crack growth were obtained by Li et al. (1988) using a physically based creep failure constitutive relation.

S39

of length scales such as inclusion sizes and spacings, and initial crack tip radii, can remove any noticeable mesh sensitivity in the numerical prediction of crack growth, provided the mesh can resolve the local stress and strain gradients at the tip of the growing crack and around the larger voids. This is true in cases where the crack growth behavior is dominated by the mechanisms governed by these length scales, but there are also cases in which other mechanisms dominate, e. g. the nucleation and growth of small scale voids, for which no length scale is specified in the formulation. DUCTILE-BRITTLE TRANSITIONS The absorbed energy versus temperature curve for the Charpy V-notch test is frequently used to characterize the brittle-ductile transition in steels, see e.g. Rolfe and Barsom (1977). The influence of strain-rate on the competing failure mechanisms in the Charpy V-notch test was analyzed by Tvergaard and Needleman (1986) for a fixed temperature, with attention focused on a temperature in the range where the transition in fracture mode between cleavage and ductile rupture takes place. In a subsequent study, Tvergaard and Needleman (1988) directly analyzed the temperature dependence of the absorbed energy for the high-nitrogen mild steel investigated by Ritchie et al. (1973). The temperature dependence of the absorbed energy results mainly from the variation with temperature of the material parameters; in particular, from the fact that the initial yield strength tends to decrease with increasing temperature, while the critical stress for cleavage is approximately constant over a wide range of temperatures. 300

Mesh 4a Mesh 4b

-ioo 0.0

0.5

2.0

Figure 6. Curves of J versus time, where J includes both an area integral term and a contour integral term, for initially blunt cracks using two different meshes. From Needleman and Tvergaard (1994). The curves of J versus time in Fig. 6 illustrate that crack initiation as well as crack growth predictions are not unduly sensitive to the discretization when there is an initially blunt crack tip. The results in Needleman and Tvergaard (1994) show that the specification

75

150 time (/isec)

225

300

Figure 7. Force versus time for three values of initial flow strength. The force is normalized by BH, where B is the Charpy specimen width and H is its thickness. From Mathur et al. (1993). The analyses in Tvergaard and Needleman (1986, 1988) were carried using a two dimensional, plane strain model of the Charpy specimen. However, the Charpy specimen is square and is not necessarily very wide compared to the dimensions of the notch region undergoing large plastic straining, so that the assumption of plane strain conditions is questionable. Mathur et al. (1993)

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

S40

MECHANICS USA 1994

Appl Mech Rev vol 47, no 6, par! 2, June 1994

carried out full three dimensional transient analyses for Charpy V-notch specimens subject to impact loading, using a d a t a parallel numerical implementation on a Connection Machine CM5 computer system. Figure 7 shows force-time curves for three values of flow strength. W i t h a0 = 175 M P a and 250 MPa, plastic yielding occurs early in the loading history and the magnitude of the initial peak increases with flow strength. Cleavage first occurs at 23 jis with CTQ =• 325 MPa, whereas when CTQ = 250 M P a cleavage does not occur until 115 fj,s. W i t h o-0 = 175 M P a the calculation was continued to 600 fis and the response remained fully ductile. Because of extensive cleavage, the forcetime curve for do = 325 M P a falls below the corresponding curves for the two lower strength materials. Also, the oscillations are larger because of less extensive plasticity. Corresponding plane strain computations were found to predict a force versus time dependence t h a t is initially in rather good agreement with the three dimensional results. However, the planar analysis cannot represent i m p o r t a n t three dimensional effects, such as the ductile failure region at the free surface edge of a crack growing mainly by the brittle mechanism, or the reduced constraint due to pull-in of the free surface in the notch region. For all three levels of flow strength the plane strain analyses predicted more rapid crack growth at the central part of the notch than found by the three dimensional analyses. .cleavage

,f=0.1

1.9

a.o

Figure 9. Crack growth versus time at various temperatures. From Tvergaard and Needleman (1993). scale associated with the cleavage model as well as the length scale of the large particle spacing, which is the main length scale associated with the ductile fracture model. SEPARATION OF SURFACES In order to analyze separation along surfaces, a theoretical framework is described where constitutive relations are specified independently for the material (or materials) and the interface. A consequence of this is t h a t a characteristic length enters the formulation and the predicted response depends on the ratio of geometric lengths to an interface characteristic length. T h e dynamic virtual work statement is written in the form / s-.SFdVJv

j Js,„,

T-8[u]dS

0 (a)

T-SudSJStX

V

'

o

(b) Figure 8. Mode of crack growth at (a) T = 273°K and (b) at T = 213°K, showing the transition between mainly ductile and mainly brittle modes of crack growth. From Tvergaard and Needleman (1993). T h e ductile-brittle transition in a plane strain edgecracked specimen subject to impulsive loading was analyzed by Tvergaard and Needleman (1993). T h e numerical results showed a clear transition between cleavage dominated crack growth at low temperatures to ductile crack growth at high temperatures. Figure 8 shows the mode of crack growth at two of the temperatures considered in Tvergaard and Needleman (1993) and Fig. 9 shows curves of crack speed versus time for all five temperatures. In these calculations there is a length

P V

W

•8udV

(6)

where u is the displacement vector, F is the deformation gradient, V and Sext are the volume and the external surface, p is the density of the material in the reference configuration, s is the non-symmetric nominal stress tensor and T is the surface traction. Also, S{nt denotes internal cohesive surface across which tractions are continuous but a displacement j u m p , [u] = u + — u~~ is permitted. T h e constitutive law for the cohesive surface is taken to be a phenomenological mechanical relation between the traction and displacement j u m p across the surface. T h e behavior t h a t needs to be captured is t h a t , as the cohesive surface separates, the m a g n i t u d e of the traction at first increases, reaches a m a x i m u m and then approaches zero with increasing separation. Various, very different, mechanisms give rise to this sort of response; for example, separation of atomic planes (Rose et al, 1983) and ductile void growth and coalescence (Tvergaard and Hutchinson, 1992). W h a t distinguishes the various mechanisms is the stress required for separation, the length scale over which the separation process

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Appl Mech Rev vol 47, no 6, part 2, June 1994

Needleman: Computational modeling of material failure

takes place and the dissipation accompanying separation. It is also worth noting that in modeling the material near the cohesive surface as a classical plastic or viscoplastic solid, t h a t stress levels are limited by plastic flow. If sufficient mobile dislocations are not available near the interface, then much higher stresses may be present, as discussed by Suo et al. (1993) in the context of fracture mechanics. Also, at a sufficiently small scale, the influence of in dividual, m a t r i x dislocation fields on the stress concentration at the interface can become i m p o r t a n t . Thus, consideration of the size scale of the separation process being modeled is needed not only for developing a constitutive relation for the cohesive surface, but also for characterizing the surrounding material. In most analyses to date, the cohesive surface constitutive relation has been taken to be elastic so t h a t any dissipation associated with separation is neglected. This framework has been used to address issues regarding void nucleation (e.g., Needleman, 1987; Tvergaard, 1990b; Povirk, et al, 1991; Xu and Needleman, 1994), quasi-static crack growth (Needleman, 1990ab; Tvergaard and Hutchinson, 1992, 1993), stability of the separation process (Suo et al, 1992; Levy, 1993), reinforcement cracking in .metal-matrix composites (Finot et al, 1993) and fast crack growth in brittle solids (Xu and Needleman, 1993).

S41

ing of a circular cylindrical rigid inclusion in an elasticviscoplastic m a t r i x . One aim of this type of analysis is develop a void nucleation criterion to use in (4). Also shown in Fig. 10 are analyses for a void and for a rigidly bonded inclusion. T h e case analyzed is one where under quasi-static conditions, decohesion requires decreasing applied displacement so there is no stable equilibrium decohesion p a t h . For both the rigidly bonded inclusion and the initial void, the oscillations d a m p out rather quickly once substantial m a t r i x plastic flow occurs. Under dynamic loading conditions, debonding occurs somewhat sooner t h a n under quasi-static loading conditions (Xu and Needleman, 1994), and is accompanied by fairly large stress oscillations. Damping is reduced as compared with the initial void and the perfectly bonded inclusion cases because decohesion is associated with elastic unloading in the matrix. Accurately predicting debonding requires a finer mesh than does a prediction of the response of a void or a rigid inclusion. As Fig. 10 shows convergence is achieved if the mesh is fine enough to resolve the local stress and deformation state at the interface. Also, convergence requirements limit the ratio of mesh size to the interface characteristic length. ACKNOWLEDGEMENT The support provided by by the Office of Naval Research through grant N00014-89-J-3054 is gratefully acknowledged. REFERENCES

- Rigidly Bonded Inclusion

\ O

m

2000

a.

j 111 i

i4* '500

Dashdot: 4x4 Dashed: 8x8 Solid 1:16x16 Solid 2: 24x24 0.10

if II II

0.15

Figure 10. Stress versus strain response for a cylindrical inclusion under dynamic loading conditions. T h e dynamic response of a rigidly bonded inclusion and of an initial void are shown for comparison purposes. From Xu and Needleman, work in progress. There are several motivations for modeling the dynamics of decohesion. One is t h a t under quasi-static loading conditions a state can be reached where there are no "nearby" stable equilibrium configurations and the system necessarily responds dynamically. Another motivation stems from interest in decohesion and void nucleation at high rates of straining. Some initial results are shown in Fig. 10 for the dynamic debonding of a plane strain analysis of debond-

Becker R, The effect of porosity distribution on failure, J. Mech. Phys. Solids, 35, 577-599 (1987). Becker R and Needleman A, Effect of yield surface curvature on necking and failure in porous plastic solids, J. Appl. Mech., 53, 491-499 (1986). Becker R, Needleman A, Richmond O and Tvergaard V, Void growth and failure in notched bars, J. Mech. Phys. Solids, 36, 317-351 (1988). Benson DJ, An analysis of void distribution effects on the dynamic growth and coalescence of voids in ductile metals, J. Mech. Phys. Solids, to be published (1993). Charalambides P and McMeeking RM, Near tip mechanics of stress induced microcracking in brittle materials, Mech. Mater., 6, 71-87 (1987). Chu CC and Needleman A, Void nucleation effects in biaxially stretched sheets, J. Engirt. Mat. Tech., 102, 249-256 (1980). Devaux J, Leblond JB, Mottet G, and Perrin G, Some new applications of damage models for ductile metals, in Advances in Fracture/Damage Models for Ductile Metals, AMD-Vol. 137, ASME, New York, NY, 181-194 (1992). Finot M, Shen Y-L, Needleman A and Suresh S, Micromechanical modelling of reinforcement fracture in particle-reinforced metal-matrix composites, to be published (1993). Gurson AL, Plastic Flow and Fracture Behavior of Ductile Materials Incorporating Void Nucleation, Growth and Interaction, Ph.D. Thesis, Brown University (1975). Hancock JW and Brown DK, On the role of strain and stress state in ductile failure, J. Mech. Phys. Solids, 3 1 , 1-24 (1983). Hancock JW and MacKenzie AC, On the mechanisms of ductile failure in high-strength steels subjected to multiaxial stress-states, J. Mech. Phys. Solids, 24, 147-169 (1976).

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

S42

MECHANICS USA 1994

Kachanov LM, T i m e of t h e r u p t u r e process under creep conditions, Izv. Akad. Nauk. USSR Otd. Tekh. Nauk., 8, 26 (1958). Knoche P and Needleman A, T h e effect of size on the ductility of dynamically loaded tensile bars, Eur. J. Mech., 1 2 , 585-601 (1993). Leblond, J B , Perrin, G and Devaux, J, Bifurcation effects in ductile metals with d a m a g e delocalization, J. Appl. Mech., to be published (1993). Lemaitre, J, Local approach of fracture, Engin. Fract. Mech., 2 5 , 523-537 (1986). Levy A J, Separation a t a circular interface u n d e r biaxial load, to be published (1993). Li FZ, Needleman A and Shih C F , Creep crack growth by grain b o u n d a r y cavitation: crack tip fields and crack growth r a t e s u n d e r transient conditions, Int. J. Fract., 3 8 , 241-273 (1988). Magnusen P E , Dubensky EM and Koss DA, T h e effect of void arrays on void linking during ductile fracture, Acta Metall. 3 6 , 1503-1509 (1988). M a t h u r K K , N e e d l e m a n A and T v e r g a a r d V, 3D analysis of failure m o d e s in t h e C h a r p y impact test, Modell. Simul. Mater. Sci. Engin., to be published (1993). Needleman A, A c o n t i n u u m model for void nucleation by inclusion debonding, J. Appl. Mech., 54, 525-531 (1987). Needleman A, An analysis of decohesion along an imperfect interface, Int. J. Fract., 4 2 , 21-40 (1990a). Needleman A, A n analysis of tensile decohesion along an interface, / . Mech. Phys. Solids, 3 8 , 289-324 (1990b). Needleman A, T h e effect of material inertia on neck develo p m e n t , in Topics in Plasticity, Yang, W . H . (ed.), A M Press, A n n A r b o r , MI, 151-160 (1991). Needleman A, Finite element studies of localization and failure in ductile metals under dynamic loading conditions, in Proceedings of the Third International Conference on Computational Plasticity: Fundamentals and Applications, Owen, D . R . J . , O n a n t e , E. and Hinton, E. (eds.), Pineridge Press, P a r t I, 521-538 (1992). Needleman A, N u t t SR, Suresh S and T v e r g a a r d V, Matrix, reinforcement a n d interfacial failure, in Fundamentals of Metal-Matrix Composites, Suresh, S., Mortensen, A. and Needleman, A. (eds.), B u t t e r w o r t h - H e i n e m a n n , 233-250 (1993). Needleman A and T v e r g a a r d V, An analysis of ductile rupt u r e in notched bars, J. Mech. Phys. Solids, 3 2 , 461-490 (1984). Needleman A and T v e r g a a r d V, An analysis of dynamic, ductile crack growth in a double edge cracked specimen, Int. J. Fract., 4 9 , 41-67 (1991a). Needleman A and T v e r g a a r d V, A numerical s t u d y of void distribution effects on dynamic, ductile crack growth, Engin. Fract. Mech., 3 8 , 157-173 (1991b). Needleman A and T v e r g a a r d V, Analyses of plastic flow localization in m e t a l s , Appl. Mech. Rev., 4 5 , S3-S18 (1992). Needleman A and T v e r g a a r d V, Mesh effects in the analysis of d y n a m i c ductile crack growth, Engin. Fract. Mech., 4 7 , 75-91 (1994). Needleman A, T v e r g a a r d V and Hutchinson J W , Void growth in plastic solids, in Topics in Fracture and Fatigue, Argon, A . S . (ed.), Springer-Verlag, New York, 145178 (1992). Ortiz M and Giannakopoulos A E , Crack propagation in monolithic ceramics under mixed m o d e loading, Int. J. Fract., 4 4 , 233-258 (1990). P a n J, Saje M and Needleman A, Localization of deformation in r a t e sensitive porous plastic solids, Int. J. Fract., 2 1 , 261-278 (1983). Povirk GL, Needleman A and N u t t SR, An analysis of t h e effect of residual stresses on void nucleation in whisker

Appl Mech Rev vol 47, no 6, part 2, June 1994

composites, Mater. Sci. Eng., A 1 3 2 , 31-38 (1991). Rice J R , Elastic-plastic fracture mechanics, in The Mechanics of Fracture, Erdogan, F . (ed.), 23-53, AMD-Vol. 19, A S M E , New York, N Y (1976). Ritchie R O , K n o t t J F and Rice JR, On t h e relationship between critical tensile stress and fracture toughness in mild steel, J. Mech. Phys. Solids, 2 1 , 395-410 (1973). Rolfe S T and B a r s o m J M Fracture and Fatigue Control in Structures - Applications of Fracture Mechanics, Prentice Hall, Englewood Cliffs NJ (1977). Rose J H , Ferrante J and Smith J, Universal binding energy curves for metals and bimetallic interfaces, Phys. Rev. Lett., 47 675-678 (1981). Suo Z, Ortiz M and Needleman A. Stability of solids with interfaces, J. Mech. Phys. Solids, 4 0 , 613-640 (1992). Suo Z, Shih, C F and Varias, AG, A theory for cleavage cracking in the presence of plastic flow, Acta. Metall. Mater., 4 1 , 1551-1557 (1993). Tong W and Ravichandran G, Inertia! effects on void growth in porous viscoplastic materials, to be published (1993). T v e r g a a r d V, Influence of voids on shear b a n d instabilities under plane strain conditions, Int. I. Fract., 1 7 , 389-407 (1981). T v e r g a a r d V, O n localization in ductile materials containing spherical voids, Int. J. Fract., 1 8 , 237-252 (1982). T v e r g a a r d V, Material failure by void growth to coalescence, Adv. Appl. Mech., 2 7 , 83-151 (1990a). T v e r g a a r d V, Effect of fibre debonding in a whiskerreinforced metal, Mater. Sci. Eng., A 1 2 5 , 203-213 (1990b). T v e r g a a r d V and Hutchinson J W , T h e relation between crack growth resistance and fracture process p a r a m e t e r s in elastic-plastic solids, J. Mech. Phys. Solids, 4 0 , 13771392 (1992). T v e r g a a r d V and Hutchinson J W , T h e influence of plasticity on mixed m o d e interface toughness. J. Mech. Phys. Solids, 4 1 , 1119-1135 (1993). T v e r g a a r d V and Needleman A, Analysis of t h e cup-cone fracture in a round tensile bar, Acta Metall., 3 2 , 157-169 (1984). T v e r g a a r d V and Needleman A, Effect of material rate sensitivity on failure m o d e s in t h e C h a r p y V-notch test, J. Mech. Phys. Solids, 3 4 , 213-241 (1986). T v e r g a a r d V and Needleman A, An analysis of t e m p e r a t u r e and r a t e dependence of C h a r p y V-notch energies for a high nitrogen steel, Int. J. Fract., 3 7 , 197-215 (1988). T v e r g a a r d V and Needleman A, Ductile failure m o d e s in dynamically loaded notched bars, in Damage Mechanics in Engineering Materials, Ju, J.W., Krajcinovic, D., Schreyer, H.L. (eds.), AMD-Vol. 109, A S M E , New York, NY, 117-128 (1990). T v e r g a a r d V and Needleman A, Effect of crack meandering on dynamic, ductile fracture, J. Mech. Phys. Solids, 4 0 , 447-471 (1992). T v e r g a a r d V and Needleman A, An analysis of t h e brittleductile transition in d y n a m i c crack growth, Int. J. Fract., 5 9 , 53-67 (1993). Xu X - P and Needleman A, Numerical simulations of fast crack growth in brittle solids, to be published (1993). Xu X - P and Needleman A, C o n t i n u u m modelling of interfacial decohesion, in Dislocations 93, Solid State Phenomena Vol. 35-36, Rabier, J., George, A., Brechet, Y. and Kubin, L. (eds.), 287-302, Scitec Publications, Switzerland (1994). Zavaliangos A and A n a n d L, Thermo-elastoviscoplasticity of isotropic porous materials, J. Mech. Phys. Solids, 4 1 , 1087-1118 (1993).

Downloaded 22 Apr 2011 to 171.67.216.21. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm