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Engineering Fracture Mechanics 75 (2008) 3276­3293 www.elsevier.com/locate/engfracmech

Ductile fracture initiation and propagation modeling using damage plasticity theory

Liang Xue *, Tomasz Wierzbicki

Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, 5-011, Cambridge, MA 02139, USA Received 30 September 2006; received in revised form 18 July 2007; accepted 28 August 2007 Available online 7 September 2007

Abstract Ductile fracture is often considered as the consequences of the accumulation of plastic damage. This paper is concerned with the application of a recently developed damage plasticity theory incorporates the pressure sensitivity and the Lode angle dependence into a nonlinear damage rule and the material deterioration. The ductile damaging process is calculated through the so-called ``cylindrical decomposition'' method. The constitutive equations are discussed and numerically implemented. An experimental and numerical investigation for three-point bending tests is reported for aluminum alloy 2024-T351. Crack initiation and propagation in compact tension specimens are also studied numerically. These simulation results show good agreement with experiments. The present model can successfully predict slant fracture as well as the formation of shear lips. Ó 2007 Elsevier Ltd. All rights reserved.

Keywords: Ductile fracture; Aluminum alloy; Damage plasticity theory; Crack propagation; Ductile rupture

1. Introduction The prediction of ductile fracture offers considerable challenges in many engineering applications. Numerous researches show that a robust algorithm for ductile materials at constitutive level is essential for solving problems involving crack formation and propagation. The solution algorithms have to be based on physical understanding of the ductile fracture phenomenon. The review of the existing experimental data in the area of fracture sheds some light on the ductile fracture modeling. The general observations from ductile fracture tests suggest that (1) superimposed hydrostatic pressure increases material ductility; (2) the plane strain (or shear) ductility is usually smaller than the ductility of in generalized tension (or generalized compression) when the pressure remains constant; (3) the ductile damage accumulates in an accelerative fashion, initially at a slower rate and then at a much faster pace towards fracture initiation; (4) the microscopic damage affects the material strength of the macroscopic mechanical

* Corresponding author. Current address: Hess Corporation, 500 Dallas Street 2250, Houston, TX 77002, USA. Tel.: +1 713 609 4326; fax: +1 713 609 5667. E-mail address: [email protected] (L. Xue).

0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.08.012

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response. Based on the above observations, a three-dimensional framework ­ the so-called ``cylindrical decomposition'' ­ for fracture modeling based on the cylindrical representation of the principal stress space has been recently established [1]. A strain based nonlinear damage rule is postulated for the ductile material. The pressure sensitivity and the Lode angle dependence are incorporated through a fracture strain envelope. The amount of damage is calculated by integrating the rate form of damage rule in which the current deformation state is compared with respect to the fracture envelope. Continuous efforts have been made in the literature on the numerical simulation of ductile fracture problems using various constitutive models. One type of damage models uses the cumulative strain damage measure, such as Johnson and Cook [2] and Wilkins et al. [3]. Most damage models in this category simply count down a so-called damage variable toward the fracture initiation strain in a linear way. Further development of this line of damage models is the continuum damage mechanics model which includes the ductile damage to the yield condition to reflect the material deterioration, e.g. Lemaitre [4]. This allows material weakening to occur and promotes localization in the weakened direction. Another type of damage models is the micromechanical models. This line of damage models often rely on the global mechanical response of some kind of microscopic representative volume structure, such as Gurson [5], Tvergaard [6], Rousselier [7] and Benzerga [8,9]. Successful simulation results have been achieved using these types of models for various cases [10­12]. The present model adopts a mathematical framework of the cylindrical decomposition that addresses the four aspects of ductile fracture mentioned above in an explicit way. It falls into the category of continuum damage mechanics. Numerical simulations are performed and two examples of well-known loading cases are given to show the applicability of the present model to practical fracture problems. In both cases, the numerically obtained fracture patterns agree with experimental observations in all main features. 2. Constitutive equations The classical plasticity theories often neglect the microscopic structural change of the materials during the loading process. On the other hand, fractographic observations show significant evolution of microstructure, which often include micro-cracks initiation and propagation and micro-voids nucleation and growth along with plastic deformation [13­15], especially in the highly deformed fracture process zone. These damages change the connectivity of the matrix materials and, therefore, the mechanical response of the material defers from the matrix material. In the present model, the strength of the matrix material is assumed to be a basic property of the material which does not change by the damage evolution of the solids. The macroscopic behavior of the solids is affected by a weakening factor which takes into account of the microscopic damage. It is further assumed that the material and the damaging process are isotropic. The plastic deformation is assumed to be non-dilatational. A von Mises type of yield condition is assumed at any damaged state and associated flow rule is adopted. 2.1. Cylindrical decomposition of damage The ductile damage accumulation is loading path dependent. Therefore, the damage is calculated in a rate form. In the present model, the damage rate is calculated through a so-called ``cylindrical decomposition'', which is adopted under the fundamental hypothesis that ``the damaging process is self-similar on any deviatorically proportional loading path at any given pressure'' [1]. The cylindrical decomposition uses the pressure, the Lode angle, the plastic strain and its rate to determine the damage rate. The pressure sensitivity and the Lode dependence of fracture are included through a fracture envelope, which looks like a ``blossom'' oriented in the hydrostatic axis in the principal stress space. The fracture envelope can be equally represented in the space of plastic strain and the mean stress, as shown in Fig. 1. The vertical axis is the mean stress and the horizontal axes are the principal plastic strain components. The intersection of the fracture envelope with a vertical plane (thick solid line) shows the pressure dependence of the material. The horizontal intersection of the fracture envelope (thick dash line) depicts the Lode angle dependence. Detailed discussion of these two functions will follow in the next two subsections. Let us consider an incremental stress from a given stress condition on the yield surface and pointing outward. From the assumption of non-dilatational plastic deformation, the hydrostatic pressure component does

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x

z

Fig. 1. The fracture surface in the three-dimensional space of the plastic strain plane and the mean stress.

not create plastic deformation. The Lode angle increment is tangent to the yield surface, which does not generate plastic deformation either. Therefore, the only factor that creates new damage is the equivalent stress increment which allows plastic flow to occur. Now, the damage can be evaluated by comparing the arbitrary loading increment to its projection on a deviatorically proportional loading path [1]. On the deviatorically proportional loading path, the pressure and the Lode angle remain constant. The self-similarity hypothesis assumes that the damaging process on all deviatorically proportional loading path can be described by a uniform damage accumulation rule. There are multiple ways of measuring the damage. In the present paper, the loss of ductility is adopted to quantify the damage in a nonlinear way. For a complex loading path, the damage is calculated in a cumulative fashion. An extreme case is the reverse loading, such as in the low-cycle-fatigue situation. It is assumed that the backward motion creates the same amount of damage as the forward motion. A direct corollary from this assumption is that the fracture loci on the plastic strain plane at a given hydrostatic pressure are symmetric to the three plane strain axes. By the assumption of isotropy, the fracture loci also have permutation symmetry with respect to the three principal axes. Therefore, the fracture loci on an plastic strain plane are identical in all twelve segments (apart from reflections). In the principal stress space, the pressure and the Lode angle are orthogonal. We assume the pressure sensitivity and the Lode angle dependence on the fracture strain are independent of each other. Combining the pressure sensitivity and the Lode dependence function by multiplication, the fracture envelope is assumed to take the form of ef ¼ ef0 lp ðpÞlv ðvÞ; ð1Þ

where ef0 is a reference fracture strain indicated by zero mean stress tension, lp(p) describes the pressure dependence and lv(v) characterizes the Lode angle dependence. The pressure and Lode angle dependence functions ­ lp(p) and lv(v) ­ are discussed in the following subsections. The shape of Eq. (1) is shown in Fig. 1. 2.2. Pressure sensitivity Solid materials show higher ductility when subjected to compressive hydrostatic pressure [18]. The pressure effect on the fracture strain is sometimes described by the stress triaxiality ratio for monotonic and close to

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proportional loading paths, such as in Refs. [19,2,20]. Microscopically, this phenomenon is due to the suppression of initiation and propagation of micro-cracks and nucleation and growth of micro-voids. It is assumed that there exists a limiting pressure above which no damage occurs. Indeed, material may even be cured for existing cracks under high hydrostatic compression. For instance, cold press welding utilizes such material behavior to create joints [21]. Based on this observation, a logarithmic function is used to describe the pressure dependence function of the fracture strain lp in Eq. (1), i.e. p lp ¼ 1 À q log 1 À ; ð2Þ plim where p = À(1/3)rii is the current hydrostatic pressure at the material point, plim is the limiting pressure beyond which no damage occurs and q is the shape parameter of the pressure sensitivity function. A cut-off pressure in the positive side of mean stress emerges from Eq. (2) when lp falls below zero, which indicates the volumetric deformation becomes important and the material shatters under lofty hydrostatic tensions. In most of the existing literature, e.g. Rice and Tracey [22], Hancock and Brown [23], Johnson and Cook [2], an exponential function with respect to the stress triaxiality was adopted. By carefully calibrating the corresponding material parameters for the proposed logarithmic function and the exponential function, the difference between the two types of pressure sensitivity functions would be small. 2.3. Lode angle dependence The Lode angle distinguishes the deviatoric state. For isotropic material, the azimuthal angle on a hydrostatic plane can be divided into six identical regions. In pffiffiffi sextant, the azimuthal angle can be characterized each by the Lode angle which is defined as tanðhL Þ ¼ ÀlL = 3, where À30° 6 hL 6 30°, lL = (2r2­r1­r3)/(r1­r3) and r1, r2 and r3 are the maximum, intermediate and minimum principal stresses [24]. Alternatively, a deviatoric state parameter v, which is defined as the relative ratio of the principal stress components, i.e. r2 À r3 ; ð3Þ v¼ r1 À r3 which varies from 0 to 1, can be used to describe the Lode angle. The Lode angle and the relative stress ratio v can be used interchangeably. The Lode angle effect on ductile fracture is often ignored in fracture modeling, such as in Johnson and Cook [2] and Gurson [5]. Experimental results show that material is more easily to break in plane strain condition than in other stress conditions on an octahedral plane [25,3]. Although there are some evidence from collective tests [3] and from sheet metal forming, e.g. reference [26], there is so far no conclusive experimental results shows the shape of the Lode dependence function. The main reason is that no device has been developed to test the material under a constant hydrostatic pressure. It should be noted that for many tests have been carried out in a pressure chamber which provides constant confining pressure, such as in Bridgman [18] and Lewandowski and Lowhaphandu [27]. However, the pressure at the fracture site varies along the loading path. A heuristic way is used to model the Lode dependence by connecting the fracture points of interval generalized tension/generalized shear/and generalized compression using a straight line on the plastic strain plane on the same hydrostatic level. This gives the formula of the first kind of Lode angle dependence function: 8 pffiffiffiffiffiffiffiffiffiffiffi v2 Àvþ1 > 0 6 v 6 0:5; > < 1þÀpffi À2Áv ; 3 c ð4Þ lv ¼ pffiffiffiffiffiffiffiffiffiffiffi > À v2 Àvþ1 > pffi Á : ; 0:5 < v 6 1; 3

c

À2 ð1ÀvÞ

where c is non-negative material constant defined as the ratio of the fracture strain between generalized shear (v = 0.5) and generalized tension (v = 0) subjectedffi to the same hydrostatic pressure. pffiffi Eq. (4) reduces to a right hexagon when c ¼ 3=2. An example of this Lode angle dependence function is shown in Fig. 2a. More generally, the Lode angle dependence function can also be assumed to take a curvilinear form, which is described by

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2

90 120 60 120

2

90 60

=

150

7 0.8 0 = .75 0 = .50 0 = .25

0 180 30 150

=

1.0 0 = .8 0 = .6 .4 .2

0 30

0 =

0 =

180

210

330

3

1

210

330

3

1

240 270

300

240 270

300

Fig. 2. Two kinds of Lode angle dependence functions.

6jhL j lh ¼ c þ ð1 À cÞ p

k ; ð5Þ

where jÆj denotes the absolute value and k is a shape parameter. This kind of Lode angle dependence function is plotted for several c values and k = 1 in Fig. 2b. The second kind of Lode angle dependence function reduces to a circle when c = 1. Xue also proposed a micro mechanical model by including the void shearing effect in Gurson-like constitutive models of porous media [44]. Similar approach was taken by Nahshon and Hutchinson [45]. In both models, the Lode angle dependence of ductile fracture are included. 2.4. Damage rule Mechanical properties of ductile solids change along with the accumulation of damage, for example the remaining ductility and the elastic stiffness both decrease if the material is plastically deformed. Therefore, it is essential to include a realistic evolution law and the consequences of damage to material strength in the constitutive equations. The damage rule describes how damage is accumulated on a constant pressure plane at a fixed azimuthal angle. Because the plastic irreversible damage is often accumulated in an arbitrary loading paths for practical problems, the damage has to be calculated in the rate form. In the damage rule, we use the ductility damage, which is defined as the relative reduction of deformability to quantify damage [1], viz. if the material can survive 10 times of the same loading, we say there is 10% ductility damage at the end of a single loading branch. In analogue to the Manson­Coffin empirical relationship and using Palmgren­Mines rule [28,29], a power law damage evolution law is assumed in the form of ðmÀ1Þ ep dep dD ¼ m ; ð6Þ ef ef where m is the damage exponent for the evolution law and ep is the current plastic strain and ef is the fracture envelope from Eq. (1). The onset of fracture is determined by the integral Z ec D¼ dD ¼ 1; ð7Þ

0

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where ec is the plastic strain at fracture on the given loading history. It should be noted that ep changes along the deformation path in general. It can also be verified that Eq. (7) yields ec = ef for any given m when ef is constant. 2.5. Material deterioration When damage accrues, material strength is decreased due to the reduction of effective load carrying area [4]. This effect can be modeled micromechanically if the damage is considered as a growing void. In a macroscopic phenomenological way, the plastic damage induced material weakening is modeled by introducing a scalar function to the matrix strength. The relative reduction of the effective load carrying area causes a loss in the elastic modulus of the material. In general, the stiffness damage does not necessarily follow the same decaying curves as the ductility damage does when the plastic deformation continues. In particular, we consider a special form of the weakening effect here. The macroscopic material strength is considered to be affected by a weakening factor of (1 À Db), i.e. req ¼ ð1 À Db ÞrM ; ð8Þ

where rM is the matrix stress­strain relationship, req is the applied equivalent stress and b is the weakening factor assumed to be a non-negative constant. For the linear case of Eq. (8) (i.e. b = 1), the stiffness damage is assumed to be the same as the ductility damage. A more careful calibration found that the stiffness damage lags behind the accumulation of ductility damage [30]. 3. Numerical examples The new ductile fracture model has been incorporated into LS­DYNA as a user defined material subroutine. Previously, successful simulations were performed for the cup­cone fracture of tensile round bar, the 45° transverse plane strain crack and the 55° plane stress crack [1]. In the present paper, further numerical studies are carried out to extend the model of crack initiation and propagation to other well-studied loading cases. Examples of tests using compact tension (CT) specimen and three-point bending (TPB) specimen will be given. 3.1. Calibration of material parameters The present damage plasticity model involves six parameters characterizing fracture. Ideally, the four parameters defining the fracture envelope, i.e. ef0, plim, q and c can be calibrated from deviatorically proportional loadings where the two stress state variables v and p are fixed. However, in reality, one or both of these two stress state variables are constantly changing along the loading paths even for simple laboratory setups. Averaging method has to be used to obtain a first estimation of these parameters. One way of finding the damage exponent m is to use existing results form low-cycle-fatigue tests. The material weakening exponent b can be found by fitting the stress­strain relationship at ranges of low plastic strains, where there is little stiffness damage. The first attempt to calibrate the present fracture parameters for aluminum alloy 2024-T351 was to make reference to a set of previous experiments by Bao and Wierzbicki [20] using a linear weakening function, i.e. b = 1. The stress­strain curve is extrapolated with a straight line from the onset of necking (see Fig. 3), which leads to a higher than conceivable matrix stress­strain curve. It is consequently found that because of the overestimated stress­strain curve and an overestimated stiffness reduction with b = 1, the overall response agrees with experiments. The calibrated material constants are listed in the second row of Table 1. A more carefully planned and comprehensive experimental and numerical program for calibration of the same material was conducted and reported in Xue [30]. Four load conditions: (1) compression of cylinder; (2) tension of smooth round bar; (3) tension of notched round bar and (4) tension of transversely doublegrooved flat plate ­ were used to calibrate the material constants. These selected test conditions cover a wide range of the pressure and the Lode angle of the stress condition. The matrix strength is calibrated using upsetting test results where little stiffness damage is to the material in the phase of uniform compression. By using a Swift type of three parameter power law relationship to fit the conventional true stress­strain curve, it was

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1000 matrix (=1.0) 900 800 matrix (=2.0) 700

Stress (MPa)

600 500 400 300 200 100 0

conventional (=)

0

0.1

0.2

0.3

0.4

0.5

0.6

True Strain

Fig. 3. The conventional true stress­strain curve and the calibrated matrix strength curves (ep­rM) for aluminum alloy 2024-T351.

Table 1 Damage plasticity material constants for aluminum alloy 2024-T351 ef0 Initial set Final set 0.70 0.80 plim 926 MPa 800 MPa q 0.97 1.5 c 0.30 0.40 m 2.04 2.0 b 1.0 2.0 Remarks CTS (Section 3.2) TPB (Section 3.3)

found that b = 2 gives a more close fit to the experimental curve in the later stage of compression. The fitted matrix stress­strain relationship is then characterized by ep 0:173 MPa: ð9Þ rM ¼ 302 1 þ 0:00387 In parallel to the laboratory tests, four numerical simulations of the same tests are run at a time to fine tune the material parameters. The numerically obtained load­displacement curves are compared with the experimental ones. The material parameters obtained after this numerical calibration procedure are listed in the third row of Table 1. For the original set (b = 1) and the final set (b = 2) of material fracture parameters, the conjugate matrix stress­strain curves are plotted in Fig. 3 along with the conventional ``true'' stress­strain curve, where no material weakening is considered (except for the abrupt decline at fracture), or in other words b = 1. 3.2. Compact tension test The compact tension (CT) specimens often develop slant cracks within certain range of thickness to ligament ratio for some materials and in other cases a flat crack is observed [31,32,46]. Simulation of the CT specimen reveals a complex crack pattern. Close to the center of the crack tip in the thickness direction, fracture initiates in Mode I flat crack. Near the lateral surface of the specimen, the through thickness stress component disappears. In the process of the crack propagation towards the surface, shear lips form in a similar way as in a cup­cone fracture due to the transition of stress state. The initially antisymmetric small shear lips create a nonsymmetric configuration at the crack tip and eventually turn the crack into a slant mode throughout the thickness direction [33,34].

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Numerical simulation efforts have been made to obtain a realistic fracture mode. It was found by many authors that the load­displacement curve at the pin can be predicted with plane strain finite element model using void-nucleation-growth-coalescence model [35]. Three-dimensional finite element model using cracktip-opening angle is also used to show the tunneling effect [36,37]. Mathur et al. [38] shows that the void volume fraction become localized in two shear bands formed in the 45° direction ahead of the crack tip for thin sheets under adiabatic condition. Besson [39] predicted a shear localization when introducing anisotropy in the Gologanu model [40]. Explicit modeling of discrete voids on the crack path is also pursued by Gao et al. [41], Tvergaard and Hutchinson [42] and Tvergaard and Needleman [43] for crack propagation. In the present study, we focus on the predicting of the slant fracture mode using the damage plasticity model. The specimen is designed according to ASTM E399 with a/W = 0.5 and a plate thickness of 0.25 in. The geometry is shown in Fig. 4. In the scope of the damage plasticity model, for all possible effects that lead to a slant crack, we restricted ourselves in the present study to (1) the Lode angle effect and (2) the weakening effect. In the constitutive model, by activating and deactivating these two effects, there are four cases of different combinations. Therefore, four calculations are run in this study for the possible combinations: Case (a) with both Lode angle dependence and material weakening effect; Case (b) with Lode angle dependence and without material weakening effect; Case (c) without Lode angle dependence and with material weakening effect and Case (d) without either Lode angle dependence or material weakening effect. The two calculations deactivating Lode dependence of fracture assumes the same fracture behavior for all deviatoric situations, i.e. lv = 1. For the runs with material deterioration, the matrix stress­strain curve with b = 1 is used; for non-deterioration runs, the conventional true stress­strain curve is the input. The fracture modes for all cases are shown in Fig. 5. From the four simulation results, it is observed that only the first simulation which considers both the material deterioration effect and the Lode dependence of fracture predicts a slant fracture mode. The crack is initially flat in the pre-crack plane and shows some tunneling effect where the crack advance in the mid-plane is the maximum. Soon after the tunnel is developed, two shear lips form in the opposite direction, about 45° to the free surfaces, and the crack is no longer flat in the entire thickness direction. When the crack propagates further, the two shear lips joint together and form a slant fracture surface over the entire thickness of the plate. A time sequence of the crack opening for the first case is shown in Fig. 6. In the remaining three simulations, a flat fracture pattern is predicted. The initial tunneling effect was captured by all four cases. It is also found that the crack propagates slower for cases without the Lode dependence

Fig. 4. The geometry configuration of the compact tension specimen.

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Fig. 5. The fracture modes of a compact tension specimen for the four cases. The von Mises equivalent stress contours are plotted.

(c) and (d) than those with Lode dependence (a) and (b). This is due to the inadequate characterization of the lower ductility in shear or plane strain condition. Mahmoud and Lease [33] conducted compact tension tests of the same material. For a larger panel with the same thickness as in the above case, they found a slant crack formed and propagated in the ligament of the specimen. The element size along the crack path is 0.33 mm · 0.28 mm · 0.32 mm (W · H · T). It is noteworthy that the mesh along the crack path was slightly skewed in the thickness direction, as shown in Fig. 7. This artificial treatment is necessary to suppress perfect symmetry because trial calculations with a symmetric mesh showed a symmetric ``X'' shaped crack developed due to the symmetry in the normal direction to the initial crack surface (vertical direction). In the finite element model, the skewed mesh is designed such that there is an approximate 0.5% difference between the two element edge lengths in the vertical direction for each element along the crack path. The same mesh is used in all four runs. The load­displacement curves for the four cases are plotted in Fig. 8. The shape of the curves from models considering the material weakening agrees well with the experimental curves obtained by Mahmoud and Lease [33]. A sharp drop in loading at the initiation of fracture for the two cases that ignore the material weakening effect is not observed in their experiments. It should be noted the width of the present numerical model is smaller than that tested by Mahmoud and Lease and, therefore, the level of load is lower. In summary, the numerical simulation results show that both the Lode dependence and the material weakening play vital roles in predicting a slant fracture in compact tension specimen. The synergistic effect of the two makes the transition from a flat crack to a slant crack possible. A dedicated paper discussing various aspects and influencing factors of the crack initiation and propagation in compact tension tests, such as the tunneling, the mode transition, the strain hardening effect and the c effect, is in preparation.

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Fig. 6. The time sequence of the flat tunneling, transition from flat to slant crack and the slant crack propagation corresponding to Fig. 5a. Plotted are the von Mises equivalent stress.

3.3. Three-point bending Three-point bending test are often used to study the fracture of ductile metals. The test setup is shown in Fig. 9. The 50 mm diameter center pin moves downward while the two smaller support pins are stationary. Rectangular cross-section beams made of 2024-T351 aluminum alloy with same height (20 mm) and three

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Fig. 7. The initial mesh and a left view showing the skew of mesh in the thickness direction.

15

Lode + softening Lode softening none

Normal Force (kN)

10

5

0

0

1

2

3

4

5

6

7

Load line Displscement (mm)

Fig. 8. Comparison of the normal load versus load­line displacement curve for the four cases.

different widths (10 mm, 30 mm and 60 mm) were loaded all the way to fracture. No pre-crack or notch is cut in the specimen. All specimens were machined from a block of metal with 4 · 4 in. cross-section. The specimen longitudinal direction is the same as the block longitudinal direction. The experimental center pin load versus deflection curves are plotted in Fig. 10 for all three width of beams. 3.3.1. Narrow rectangular beam (three-dimensional) In all three-point bending experiments, the fracture initiates on the tensile side of the bending beam. The cracked specimen for a narrow rectangular beam (10 mm wide) is shown in Fig. 11.

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mm

3287

20mm

240mm

mm

180mm

mm

Fig. 9. Test setup of a three-point bending test.

TPB experiments

100 90 80 70 60

60mm

Force ( kN)

30mm

50 40 30 20 10 0 0 10 20 30 40 50 60

10mm

Deflection (mm)

Fig. 10. The load­displacement curves for the rectangular beams with three different width.

Fig. 11. Fracture surface of a narrow beam under three-point bending test.

The fracture surface shown in Fig. 11 can be divided into three parts that failed by different mechanisms. Initially, the fracture initiates at the center of tensile surface that is a plane strain shear fracture, which is

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identified as a central shear zone. The fracture surface of this shear zone is relatively smooth and approximately 45° with the surface. Then, the crack propagates through the thickness direction. As the crack opens up, the beam creates new surface and the tensile stress is maximum at the center of the crack tip both due to the stress concentration and due to the material weakening. A coarse surface is observed in the main center region, which indicates that a tensile type of failure has occurred. This is indicated by a central tensile zone in Fig. 11. When the crack propagates towards the free surface in the lateral direction, shear lips develop on both sides. A full three-dimension simulation for the three-point bending of a narrow beam is carried out because of the three-dimensional fracture in nature. The symmetry condition in the width direction is utilized to reduce the computational time. The material parameters in the third row of Table 1 is used in the simulation. The fracture specimen in the simulation is shown in Fig. 12. Two friction conditions were assumed for the contact between the punch and the beam. The friction coefficient was set to 0.2 and 0.5, respectively. The crack patterns were very close for the two friction coefficient cases. The load­deflection curves at the center pin for these two conditions and the experimental measurement are plotted together in Fig. 13. It is shown that the bending resistance of the rectangular beam increases with higher friction coefficient. The simulation results shows the shear lips develop when cracks propagate towards the edges. 3.3.2. Wide rectangular bar The same fracture pattern as in the narrower bar is observed in the wider rectangular bars. Fig. 14 shows the fracture surface of a wide rectangular beam (60 mm) fractured in three-point bending test. The central shear zone of the 60 mm wide beam is more than 80% of the total width. We consider the lateral constraint is well built up in the central region. Plane strain elements are used to simulate the middle section of wide beam.

Fig. 12. Fracture surface of a narrow beam predicted by using present model. The shear lips are visible at the edges. The plotted contours are the plastic strain.

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TPB experiments (10mm wide)

16

Experiment

14

12

Simulation, =0.2

10

Force (kN)

Simulation, =0.5

8

6

4

2

0 0 10 20 30 40 50 60

Displacement (mm)

Fig. 13. The comparison of the simulation load­displacement curves with the experimental curves for the 10 mm · 20 mm (W · H) crosssection three point bending specimen.

Fig. 14. Fracture surface of a wide beam showing three types of fracture surface: central shear zone, central tensile zone and shear lips at the edges.

Similar to the compact tension case, by activating and deactivating these two effects, there are four different combinations. The predicted fracture modes of these four cases are shown in Fig. 15.

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Fig. 15. Simulation fracture pattern of a wide beam under plane strain condition under four cases of constitutive consideration.

It is shown that the central shear zone is correctly predicted if the material weakening effect is included in the constitutive model. On the other hand, a straight crack is predicted instead of a slant one if a monotonic

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TPB experiments

100 90 80 70 60

=1.0, =

Force (kN)

50 40 30 20 10 0 0 10 20 30 40 50 60

=0.4, =2.0

=0.4, = =1.0, =2.0

Displacement (mm)

Fig. 16. The predicted load­displacement curves for the four cases.

stress­strain curve is used and no weakening effect is taken into account. For the finite element model where neither the Lode angle effect nor the weakening effect is considered, the beam does not fail rather slide down between the two support pins. The plane strain condition corresponds to an infinitely wide beam in bending. Assuming the entire width of the 60 mm wide beam is under plane strain condition, the load­deflection curves at center pin are plotted for all four cases in Fig. 16. It is shown in Fig. 16 that the predicted curve by considering the calibrated c and b (case a) fractures before the experimentally observed one. From the experimental trend as shown in Fig. 10, we see the displacement continuously decay with the increase of the width of the beam. For the maximum width tested (60 mm) here, the width to thickness ratio is 3.0. A plane strain condition is not yet reached at this ratio. In fact, at a late stage of deformation, the wide beam shows a saddled shape in the tensile side of the beam. The concave curve of the tensile side of the center cross-section can be seen in the fracture specimen in Fig. 14a. This explains why using plane strain elements under predicts the pin displacement at fracture for the 60 mm wide beam.

4. Conclusions A generic formulation of a damage plasticity model for ductile fracture was recently developed. This model includes material deterioration, pressure sensitivity, Lode angle dependence and nonlinear damage evolution law. The constitutive equations are briefly presented. The focus of the present paper is on the crack initiation and propagation using the examples of compact tension specimen and three-point bending of rectangular bars. Aluminum alloy 2024-T351 is used in the present study. In the compact tension case, a slant fracture pattern is predicted. The simulation results show a slant crack pattern for a 0.25 in. thick specimen, which agrees with experimental observation. From the simulation results, it is identified that a key influencing factors for predicting a slant fracture mode is the synergistic effect of the material weakening and the Lode angle dependence of fracture strain. In the three-point bending case, rectangular bars with three different widths are tested. A combination of different fracture modes is observed on the fracture surface in the experiments. Post-mortem examination reveals that the fracture surface includes both shear dominated and tension dominated fracture zones. The complex fracture surface is captured in the numerical simulation using the damage plasticity model. For the narrow beam, a three-dimensional simulation shows the center pin load­deflection curve agrees with

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the experimental measurement. For the wider beam, a two-dimensional plane strain simulation shows the beam is not wide enough to satisfy a plane strain load condition. References

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