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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Modeling the ductile fracture behavior of an aluminum alloy 5083-H116 including the residual stress effect Jun Zhou a, Xiaosheng Gao a,⇑, Matthew Hayden b, James A. Joyce c a

Department of Mechanical Engineering, The University of Akron, Akron, OH 44325, United States Alloy Development and Mechanics Branch, Naval Surface Warfare Center, West Bethesda, MD 20817, United States c Department of Mechanical Engineering, United States Naval Academy, Annapolis, MD 21402, United States b

a r t i c l e

i n f o

Article history: Received 2 November 2011 Received in revised form 8 February 2012 Accepted 19 February 2012

Keywords: Stress triaxiality Lode angle Damage accumulation Ductile crack initiation and growth Residual stress

a b s t r a c t In this work, the plasticity and ductile fracture behaviors of an aluminum alloy 5083-H116 are studied through a series of experiments and ﬁnite element analyses. A recently developed stress state dependent plasticity model, the I1–J2–J3 plasticity model, is implemented to describe the plastic response of this material. Furthermore, a ductile failure criterion based on a damage parameter deﬁned in terms of the accumulative plastic strain as a function of the stress triaxiality and the Lode angle is established. The calibrated I1–J2–J3 plasticity model and ductile failure model are utilized to study the residual stress effect on ductile fracture resistance. A local out-of-plane compression approach is employed to generate residual stress ﬁelds in the compact tension specimens. Fracture tests of C(T) specimens having zero, positive and negative residual stresses are conducted. The numerical results, such as load–displacement curves and crack front proﬁles, are compared with experimental measurements and good agreements are observed. Both experimental and ﬁnite element results show signiﬁcant effect of residual stress on ductile fracture resistance. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Engineering structures often experience plastic deformation under normal or extreme loading conditions, leading to the loss of load carrying capacity due to ductile failure. The most popular continuum plasticity model is the so-called J2-ﬂow theory. This theory assumes hydrostatic stress as well as the third invariant of the stress deviator has no effect on plastic yielding and the ﬂow stress and it is widely employed to describe the plastic response of metals. However, increasing experimental evidences show that this assumption is invalid for many materials. Inspired by the extensive experimental results reported by Spitzig et al. [1], Brünig [2] presented an I1–J2 yield criterion which incorporate the pressure dependency. Later, the third invariant of stress deviator, J3, was added to the I1–J2 yield criterion by Brünig et al. [3] to study the deformation and localization behavior of hydrostatic stress sensitive metals. Recently, Bai and Wierzbicki [4] presented a pressure and Lode dependent plasticity model and employed this model in ductile fracture analysis of several metals; Gao et al. [5,6] noticed the plastic response of a 5083 aluminum alloy is stress-state dependent and proposed an I1–J2–J3 dependent plasticity model for this material. The I1–J2–J3 dependent plasticity model proposed by Gao et al. [6] is adopted in this study. Ductile damage models for metals mainly fall into two categories. One is based on the micromechanical theory, which considers the void nucleation, growth and coalescence mechanisms in materials. The other is macroscopic based, where

⇑ Corresponding author. Tel.: +1 330 972 2415; fax: +1 330 972 6027. E-mail address: [email protected] (X. Gao). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2012.02.014

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Nomenclature C(T) D F G

rij

r0ij

I1 J2 J3 T⁄ n ep

ef LOPC

compact tension specimen damage parameter yield function ﬂow potential stress tensor stress deviator tensor ﬁrst invariant of the stress tensor second invariant of the stress deviator tensor third invariant of the stress deviator tensor stress triaxiality Lode parameter equivalent plastic strain failure strain local out-of-plane compression

the damage criterion is often deﬁned by macro variables such as stress and strain. Most of the currently existing ductile fracture models have incorporated the stress triaxiality effect, i.e., material ductility decreases with increasing stress triaxiality. However, as demonstrated by Kim et al. [7,8] and Gao et al.[9,10], stress triaxiality alone cannot completely determine the effect of the stress state on ductile material failure, especially for the low triaxiality and shear dominated cases. Xue and Wierzbicki [11] presented a fracture model use ‘‘cylindrical decomposition’’ method which assumes stress triaxiality and Lode angle dependence on fracture strain are independent. Wierzbicki et al. [12] evaluated seven commonly used macroscopic fracture models and compared fracture loci for all seven cases constructed in the space of the equivalent fracture strain and the stress triaxiality under the plane stress condition. In this study, a damage parameter is deﬁned as a weighted integral with respect to the effective strain, where the integrand is the reciprocal of the effective failure strain as a function of the stress triaxiality and the Lode angle (related to the third invariant of the deviatoric stress tensor). The I1–J2–J3 dependent plasticity model and the stress state dependent ductile failure model are calibrated for an aluminum alloy 5083-H116 and the calibrated material models are implemented in ABAQUS/Explicit [13] via a user deﬁned subroutine VUMAT. We decide to use the explicit code because it has the ‘‘element deletion’’ function so that crack propagation can be conveniently simulated. The second part of this investigation is to examine the effects of residual stress on the ductile fracture behavior of the 5083 aluminum plate. Residual stresses in the engineering structures are generated from forming, welding, heat treatment, etc., which play an important role in either increasing or decreasing the fracture resistance. The compressive residual stress usually improves the fracture toughness, while the tensile residual stress can detrimentally reduce the loading capacity of the structure. This is usually attribute to the additional crack driving force and change of the crack front constraint [14,15]. To quantify the effect of residual stress on fracture toughness, it is necessary to introduce well characterized and reproducible residual stress ﬁelds into fracture specimens. There are plenty of literatures on residual stress generation techniques, which can either be mechanical or thermal process. Almer et al. [16] deformed large tensile specimens and cut the gauge sections to produce C(T) specimens. Meith et al. [17] applied local compression to the sides of fracture specimens. Because of the strain incompatibility between elastic and plastic region caused by the permanent plastic deformation, the residual stress ﬁeld can be generated in the specimen. This local out-of-plane compression (LOPC) approach is further explored by Mahmoudi et al. [18], who ran a series experiments and ﬁnite element analyses to examine how the position of compression tools inﬂuences the residual stress distribution in the specimen. Here we employ the LOPC approach and use two pairs of compression punches to generate various residual stress ﬁelds in C(T) specimens. The residual stress ﬁeld is obtained by conducting ﬁnite element simulation of the out-of-plane compression process. After the residual stress ﬁeld is generated, fracture tests of C(T) specimens having positive and negative residual stresses are conducted and simulated numerically. The numerical results, such as load–displacement curves and crack front proﬁle, are compared with the experiment and good agreements are observed, suggesting that the numerical model is capable of capturing the residual stress effect on ductile fracture resistance. 2. Plasticity and fracture models 2.1. Plasticity modeling The isotropic, stress state dependent plasticity model is formulated in terms of the invariants of stress tensor. Let rij be the stress tensor and r1, r2 and r3 be the principal stress values. I1 represents the ﬁrst invariant of the stress tensor and the summation convention is adopted for repeated indices. The hydrostatic stress (or mean stress) can be expressed as

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1 3

1 3

1 3

rh ¼ I1 ¼ rii ¼ ðr1 þ r2 þ r3 Þ Let

ð1Þ

r0ij be the stress deviator tensor and r01 , r02 and r03 be its principal values, i.e.

r0ij ¼ rij rh dij

ð2Þ

where dij represents the Kronecker delta. It is obvious that the ﬁrst invariant of the stress deviator tensor is zero. The second and third invariants of the stress deviator tensor are deﬁned as

h i J 2 ¼ 12 r0ij r0ji ¼ ðr01 r02 þ r02 r03 þ r03 r01 Þ ¼ 16 ðr1 r2 Þ2 þ ðr2 r3 Þ2 þ ðr3 r1 Þ2 ;

ð3Þ

J 3 ¼ detðr0ij Þ ¼ 13 r0ij r0jk r0ki ¼ r01 r02 r03

For isotropic materials, the general forms of the yield function (F) and the ﬂow potential (G) are expressed as functions of I1, J2 and J3. Eq. (4) describes the yield condition and the ﬂow rule

¼ 0; FðI1 ; J 2 ; J 3 Þ r

e_ pij ¼ k_

@GðI1 ; J 2 ; J 3 Þ @ rij

ð4Þ

is the hardening parameter, e_ pij are the rates of the plastic strain components and k_ is a positive scalar called the where r plastic multiplier. The following ﬁrst order homogeneous function is used to deﬁning the yield function (F) [6]

F ¼ c1 ða1 I61 þ 27J 32 þ b1 J 23 Þ1=6

ð5Þ

where a1, b1 are material constants and c1 is determined by substituting the uniaxial condition into Eq. (5) so that the equivalent stress deﬁned by re = F equals to the applied stress

c1 ¼ 1=ða1 þ 4b1 =729 þ 1Þ1=6

ð6Þ

The ﬂow potential (G) takes a similar form

G ¼ c2 ða2 I61 þ 27J 32 þ b2 J 23 Þ1=6 ;

c2 ¼ 1=ða2 þ 4b2 =729 þ 1Þ1=6

ð7Þ

If the ﬂow potential and the yield function are identical, i.e., F = G, a material is said to follow the associated ﬂow rule. Furthermore, if a1 = b1 = a2 = b2 = 0, the plasticity model degenerates to the formulation of classical J2-ﬂow theory and re becomes the von Mises equivalent stress. The hardening parameter depends on the strain history. By enforcing the equivalence of plastic work, i.e.,

r e_ p ¼ rij e_ pij

ð8Þ

the equivalent plastic strain increment can be deﬁned as

e_ p ¼ rij e_ pij =r

ð9Þ

ðep Þ, where ep ¼ Therefore, the hardening behavior can be described by a stress vs. plastic strain relation r

R

e_ p dt.

2.2. Fracture criterion The cumulative strain damage models assume that the damage toward eventual fracture is due to the plastic deformation history and the equivalent fracture strain depends on the stress state subjected by the material. Here a damage parameter, D, is introduced and given by

D¼

Z

ep 0

dep ef ðrÞ

ð10Þ

with ef being the failure strain under the current stress state characterized by two parameters T⁄ and n deﬁned as

T ¼

rh 27J 3 ; n¼ re 2r3e

ð11Þ

where T⁄ is the stress triaxiality ratio and n is related to the Lode angle, h, through n = cos (3h + p/2). Therefore

ef ðrÞ ¼ ef ðT ; nÞ

ð12Þ ⁄

Under proportional loading and if T and n remain unchanged during the loading history, when the equivalent plastic strain, ep , reaches the critical value ef, D will equal to unit. For general cases, when the cumulative damage according to Eq. (10) reaches one, ductile failure is said to have happened.

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In Eq. (12), if n is a constant, ef becomes a function of T⁄ only. It is well documented that the ductility of metals increases when the material is subjected to hydrostatic pressure. Here an exponentially decaying function having the same form as the Johnson–Cook fracture model [19] is used to describe the dependency of ef on T⁄

en¼const: ¼ ½A þ B expðC T Þ f

ð13Þ

where A, B and C are material constants to be calibrated using experimental data. The Lode angle distinguishes the deviatoric stress state and it is mathematically convenient to use the parameter n deﬁned in Eq. (11), whose range is from 1 to 1, to quantify the Lode angle. Wilkins et al. [20] was ﬁrst to introduce the effect of Lode angle on ductile fracture, where the function ef(T⁄, n) was taken to be symmetric with respect to n. Here we follow Xue and Wierzbicki [21] and assume ef(T⁄, n) take the following form

ef ðT ; nÞ ¼ en¼1 ðT Þ ðen¼1 ðT Þ en¼0 ðT ÞÞ½1 ð1 jnjn Þ1=n f f f

ð14Þ

where a symmetric function of n is used to interpolate the value of ef between two bounding values efn¼1 and efn¼0 . The two bounding curves, en¼1 ðT Þ and en¼0 ðT Þ given by Eq. (15), can be calibrated by conducting simple mechanical tests: n = 1 for f f notched, round tensile specimens and n = 0 for ﬂat-grooved plates under tension and the thin-walled torsion specimen. Calibration of parameter n requires performing additional tests using specimens having n values between 0 and 1, which can be done by conducting combined torsion–tension tests of thin-walled cylindrical specimens.

en¼1 ¼ ½A1 þ B1 expðC 1 T Þ; en¼0 ¼ ½A2 þ B2 expðC 2 T Þ f f

ð15Þ

Fracture is assumed to have initiated at a material point once the failure criterion is reached. The post-initiation softening process needs to be considered in order to model crack propagation. As illustrated by Li et al. [22], because the ﬁnite element has a ﬁnite size, additional work is needed to propagate the crack through the element, i.e., the element gradually loses its strength as crack grows through it. A mesh-independent, post-initiation material degradation model based on an effective plastic displacement (uf) is available in ABAQUS [13] and is adopted in this study. The element is removed when it is fully degraded (stresses being reduced to zero). The plasticity and ductile fracture models described above are implemented into ABAQUS/Explicit via a user deﬁned subroutine VUMAT following the procedures developed by Kim and Gao [23] and Gao et al. [6]. The explicit code is used here because it has the ‘‘element deletion’’ function so that crack propagation can be conveniently simulated. 3. Model calibration for aluminum alloy 5083-H116 The plasticity and ductile fracture models described above are calibrated in this section for an aluminum alloy 5083H116. All specimens are machined from a 25 mm thick plate, with tensile axes oriented transversely to the rolling direction. Round specimens were turned with low stress machining procedures and rectangular specimens were electro-discharge machined. All tests are performed at room temperature and are considered to be quasi-static. The test matrix includes smooth and notched round tensile bars, cylindrical compression specimens, grooved plane strain specimens and the Lindholm-type specimen subjected to different tension–torsion ratios, as plotted in Fig. 1. Detailed descriptions of specimen geometries and ﬁnite element modeling can be found in [6]. 3.1. Calibration of plasticity model The equivalent stress–strain relation (true stress vs. logarithm strain) is obtained by the uniaxial tension test, which can be described by a power-law hardening relation

e ¼ rE when r 6 r0 e ¼ rE0 ðrr0 Þ1=N when r > r0

ð16Þ

Fig. 1. Sketches of a smooth round bar, a notched round bar, a cylindrical compression specimen, a grooved plane strain specimen and a torsion specimen.

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107

Fig. 2. Comparison of the numerical and experimental load vs. displacement curves for (a) the smooth tensile specimen and (b) the compression specimen.

with E = 68.4 GPa, r0 = 198.6 MPa, v = 0.3, and N = 0.155. The pressure dependent parameters can be calibrated by considering cylindrical compression specimens and uniaxial tensile specimens. Fig. 2 compares the numerical predictions and the experimental measurements for the smooth tensile specimen and the compression specimen respectively. The experimental data are represented by black thinner lines while the ﬁnite element result is represented by a red1 thicker line. Using the stress strain data obtained from uniaxial tensile test, the predicted load–displacement curve of the cylindrical compression specimen ﬁts well with the test data as shown in Fig. 2b, which indicates the plasticity behavior of this material has no pressure dependency, thus a1 and a2 are set to be zeros. Torsion and tension specimens exhibit different Lode parameters, thus can be used to examine the J3 dependency. As found in Ref. [6], the simulation of pure torsion test using stress–strain curve from uniaxial tensile test and the J2-plasticity model over-predicted the torque vs. twist angle responses, as shown in Fig. 3a (dotted red line), which indicates the plastic response is J3 dependent. It is found that parameter b1 has strong effect on the predicted torque vs. twist angle response of the torsion specimen while parameter b2 has pronounced effect on the predicted axial force–displacement response of torsion–tension specimen. The ﬁnal calibrated material constants are a1 = a2 = 0, b1 = 60.75 and b2 = 50. Fig. 3b and c compares the predicted and measured torque vs. twist angle and axial force–displacement responses. With these material constants determined, the yield surface (F) and ﬂow potential (G) in the deviatoric stress plane are given in Fig. 4. Other specimens which are not used in the calibration process are simulated to verify the calibrated material model. Fig. 5a compares the predicted and measure load–displacement curves for a notched round bar, Fig. 5b compares the predicted and measure load–displacement curves for a grooved plane strain specimen, and Fig. 5c and d compares the predicted and measured torque vs. twist angle and axial force–displacement responses for a torsion–tension specimen with a different torsion–tension ratio than the specimen shown in Fig. 3. As can be seen in Fig. 5, excellent agreements between numerical and experimental results are found for all cases. 3.2. Calibration of the fracture model As described in Section 2.2, failure strain is a function of T⁄ and n. However, since both T⁄ and n at a material point may vary during the loading history [6], the average values, given by Eq. (17), will be used.

T av ¼

1 ef

Z

ef 0

T dep ;

nav ¼

1 ef

Z

ef

ndep

ð17Þ

0

Using the calibrated I1–J2–J3 plasticity model, the load–displacement response of each tested specimen can be computed. The sudden load drop on the measured load–displacement curve designates fracture initiation. By comparing the computed and measured load–displacement curves, the increment number when fracture initiates can be determined. The histories of stress triaxiality, Lode parameter, and plastic strain increment of the critical element are output and T av and nav are calculated according to Eq. (17). For axi-symmetric tensile specimens, including smooth round bar and all notched round bars, n in the center element equals to 1 and T⁄ varies with the notch radius. For grooved plane strain specimens, n in the center element equals to 0 and T⁄ varies with the groove radius. For the Lindholm-type specimen subjected to pure torsion, both n and T⁄ equal to zero in the gage section. For the Lindholm-type specimen subjected to combined torsion and tension, n and T⁄ in the gage section vary with the imposed torsion–tension ratio. Fitting the failure strain vs. triaxiality data obtained from n = 1 and n = 0 specimens to functions given by Eq. (15), parameters (A1, B1, C1) and (A2, B2, C2) can be calibrated. The calibrated values are A1 = 0, B1 = 0.85, C1 = 1.9, A2 = 0, B2 = 0.64 and 1

For interpretation of color in Figs. 2–7, 9, and 11–20, the reader is referred to the web version of this article.

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TT-16

TT-16

Fig. 3. Comparisons of load vs. displacement and/or torque vs. twist angle responses between the experimental data: (a) the pure torsion specimen, (b) and (c) the torsion–tension specimen (TT-16).

Fig. 4. Yield surface and ﬂow potential in the deviatoric plane.

C2 = 1.9. The ef vs. T⁄ curves for these two cases are plotted in Fig. 6. As can be seen, for both cases the failure strain decreases dramatically as the stress triaxiality increases. The difference between the two curves indicates the Lode angle dependency, although this dependency is not as strong compared to the result obtained by Bai and Wierzbicki [4] for aluminum alloy 2024-T351. Using the two ef vs. T⁄ curves and the torsion–tension data, which have n values between 0 and 1, the shape parameters n can be determined. The orange circular symbol in Fig. 6 shows a data point obtained from a torsion– tension specimen having n = 0.81 and T⁄ = 0.31. The calibrated shape parameter for this material is n = 1.31. After the parameters in Eq. (14) are calibrated, the 3D failure surface in the space of n and T⁄ is fully deﬁned. Fig. 7 shows the 3D failure locus by using the calibrated parameters A1 = 0, B1 = 0.85, C1 = -1.9, A2 = 0, B2 = 0.64, C2 = 1.9, and n = 1.31. 4. Fracture specimen and residual stress generation The 1/2T C(T) specimens (thickness: 12.7 mm) are used in fracture tests. The speciﬁc design utilized in this study is modiﬁed from the standard design speciﬁed by ASTM-E1820 as a means of applying controlled residual stresses. The aim is to introduce residual stresses into the C(T) specimen by compressing the two faces with cylindrical punches to a speciﬁed

J. Zhou et al. / Engineering Fracture Mechanics 85 (2012) 103–116

E-Notch

TT-9

109

G-Groove

TT-9

Fig. 5. Comparisons of the predicted load vs. displacement and/or toque vs. twist angle responses: (a) notched round bar (E-Notch); (b) plane strain specimen (G-Groove); (c) and (d) torsion–tension specimen (TT-9).

Fig. 6. Failure strain vs. stress triaxiality.

displacement to produce a pair of permanent depressions on both faces of the specimen. Fig. 8 shows the geometry of the C(T) specimen. 4.1. Experiment design From the analyses performed by Mahmoudi et al. [18], the size and position of the punching tools have strong inﬂuence on the magnitude and distribution of the residual stress ﬁeld. Thus one can tailor the residual stress ﬁeld at the crack tip by varying the position of the set of depressions relative to the crack tip. Mahmoudi et al. [18] conduct two series of tests, one involves single pair of compression tools which place punches directly along the line of crack, the other one uses double pairs of punches. To avoid the direct contact of punches on the material in the crack growth plane, only double pairs punch LOPC method was adopted here. The position of punches are illustrate in Fig. 9. When the punches are ahead of the crack tip, Fig. 9a, positive residual stress will be generated; and negative residual stress will be generate for punches behind of the crack tip, Fig. 9b.

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εf

ξ T

*

Fig. 7. 3D plot of the failure locus.

Fig. 8. Geometry of the plane-sided C(T) specimen (mm).

Fig. 9. Punching tool positions.

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111

A series ﬁnite element analyses are conducted ﬁrst to determine the diameter, applied displacement, and locations of each depression, in order to obtain larger residual stress inﬂuenced region. From these analyses, the 8.89 mm punch radius is selected and the punch position is chosen to be dx/R = 1 and 1, dy/R = 1.2, where dx and dy are the distances from the punch center to the crack tip parallel and normal to the crack growth direction respectively and R is the radius of the punch. 4.2. Mechanical test preparation With the aid of the above analyses, ﬁxtures were designed for use in side compression. The specimen is sandwiched between two guide plates. The guide plates are aligned with the specimen by threading two pins through one guide plate, the pin loading holes of the specimen, and through the other guide plate. These guide pins ensure consistency in locating the side compression indentations between specimens. Two side compression punches (one on either side of the crack plane) are placed in the top guide plate and two mating punches are placed in the bottom guide plate. Once assembled, the compression force is applied to the top punches while the bottom punches remain ﬁxed. Sets of ﬁxtures were designed and machined for both the dx/R = 1 and dx/R = 1 conﬁgurations (dy/R = 1.2). Fig. 10 illustrates the schematic of side compression ﬁxture. 5. Experimental and numerical results 5.1. C(T) specimens without residual stress The plane-sided C(T) specimen is considered ﬁrst. Due to symmetry of the geometry and the boundary conditions, only a quarter of the specimen is meshed. The element size along the crack path is 0.254 mm in all three directions. The eight-node, isoparametric, brick elements with reduced integration are used in the analysis. Fig. 11a shows the quarter-symmetric ﬁnite element mesh and Fig. 11b shows a close-up of the crack tip region. Fig. 12 compares the predicted crack proﬁle with the fracture surface of the broken specimen and the agreement is very good. A ‘‘V’’ shape crack proﬁle, seen in both test and simulation results, shows strong crack tunneling effect. This is due to the variation of the constraint in the thickness direction, where the plane stress prevails at the free surface and the plane strain condition prevails in the center. Fig. 12b displays the stress triaxiality contour on the crack plane, showing the stress triaxiality decreases from the mid-plane to the free surface. The computed and measured load vs. load line displacement curves of the plane-sided C(T) specimen are compared in Fig. 13. Good agreement is observed before fracture initiation and at the early stage of crack growth. The simulation result slightly under predicts the applied force at the later stage. In order to promote plane strain constraint along the crack front and obtain more uniform though thickness crack growth, the C(T) specimens are side-grooved by 20% of the thickness (10% each side). Consequently, a quarter-symmetric ﬁnite element model is generated for the side-grooved specimen, in which the same element type and size are used as those for the plane-sided specimen. With the side grooves, the constraint level is signiﬁcantly raised close to specimen edges and as a result, more uniform crack growth (less tunneling) is observed. Fig. 14a compares the predicted crack proﬁle with the crack surface of the broken specimen. Fig. 14b shows stress triaxiality becomes almost uniform through specimen thickness due to the side grooves. Because the constraint level is raised by the side grooves, fracture becomes easier, resulting in a lower load carrying capacity by the specimen. Fig. 13 compares the load–displacement curves for the side-grooved specimen with that of the

Fig. 10. Schematic of side compression ﬁxture.

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Fig. 11. Finite element mesh of the plane-side CT specimen.

Fig. 12. Fracture surface and stress triaxiality distribution in the plane-sided C(T) specimen.

Fig. 13. Comparison of the load–displacement curves for side-grooved and plane-sided specimens.

plane-sided specimen. The load carrying capacity of the C(T) specimen is signiﬁcantly reduced by the side grooves. The model predicted load–displacement curves are also included in Fig. 13, showing good agreements with test data. In conducting the experiments, the test was stopped after some amount of crack extension and the specimen was broken by fatigue loading. The post fatigue marks were used to determine the amount of crack growth. Due to the non-uniform crack growth through the specimen thickness, a nine-point average was used to determine the crack length. We compare the model predicted crack growth with experimental measurements at the same applied load levels and ﬁnd very good agreements between the two. For examples, C(T)-17 has crack extensions of 6.81 mm (measured from post fatigue marks)

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113

Fig. 14. Fracture surface and stress triaxiality distribution in the side-grooved C(T) specimen.

when the test was stopped. The ﬁnite element analyses predict the amount of crack extension of 7.11 mm at the same load level. 5.2. Fracture tests with residual stress The C(T) specimens considered hereafter are all side-grooved. In the ﬁnite element analysis, the cylindrical punches are modeled as rigid surfaces, and a friction coefﬁcient of 0.001 (almost frictionless) between the punch and the specimen surface is used. The rigid punch is given a displacement slowly normal to the side surface of the C(T) specimen and after the reaction force reaches the level of the applied load, the punch is removed from the specimen surface. 5.2.1. Tensile residual stress Two levels of compression forces, 182 kN and 220 kN, were used in the experiments to generate tensile residual stresses. The average total indentation depths (after the punches were removed) are 0.083 mm and 0.244 mm for 182 kN and 220 kN respectively. The ﬁnite element analyses results in 0.089 mm and 0.259 mm total indentation depths for these two cases. Since the LOPC method creates residual stress ﬁeld by introducing plastic strain into structure, too much side compression may results in the crack extension. Several methods, such as dye injection and SEM observation of the fracture surface, are used to verify if this has happened. From result of the dye and also SEM observations, the tensile residual side compression does result in crack initiation and growth when 220 kN load was applied. SEM observations also showed clear damage i.e. inclusions broken and pulled loose from the matrix. However, this was not observed for the 182 kN case. The simulation results for both cases conﬁrm the experimental observations. Fig. 15 shows the crack front region after side compression, suggesting crack does extend about 1.5 mm when 220 kN of compression force is employed, whereas crack extension does not happen when 182 kN compression force is applied. The contours of residual stresses normal to the crack plane for both cases are shown in Fig. 15. The high positive residual stress is conﬁned in a small region close to the crack tip, within about 1.3 mm from the crack tip, and the residual stress distribution is fairly uniform along the crack tip except in the region close to the free surface. Fig. 16 displays the variation of the residual stress (r22) ahead of the crack front, showing that the residual stress decays as the distance from crack tip

Fig. 15. Contour plots of the residual stress normal to the crack plane: (a) 220 kN side-compression and (b) 182 kN side-compression (stress unit in psi).

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Fig. 16. Residual stresses (r22) distributions ahead of the crack tip at the mid-plane and the specimen edge.

Fig. 17. Triaxiality distribution under 182 kN LOPC (tensile residual stress).

Fig. 18. Comparisons of the computed and measured load–displacement curves for C(T) specimen without residual stress and with tensile residual stress.

increases. As can be seen, 220 kN side compression generates only slightly larger residual stress than 182 kN side compression does and the residual stress at the mid-plane is larger than at the specimen edge. Since the purpose is to study the residual stress effect on the fracture toughness, appropriate values of the compression force should be used so that the unwanted crack growth during side the compression process is avoided. Tensile residual stress not only raises the crack driving force but also introduce additional crack-tip constraint [24,25]. Fig. 17a shows the distribution of triaxiality in the crack front region under 182 kN LOPC force and Fig. 17b shows the variation of the triaxiality ahead of the crack at the mid-plane. After the residual stress ﬁelds have been generated in the specimens, ﬁnite element analyses of the compact tension tests of these specimens are carried out. Fig. 18 compares the model predicted load–displacement curves with experimental records for the as-received specimen as well as the specimens with tensile residual stress ﬁeld. The numerical model captures the effect of the tensile residual stress on the fracture resistance. The existence of tensile residual stress drastically reduces

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(a)

115

(b)

(c) Fig. 19. (a) Contour plot of the residual stress normal to the crack plane; (b) and (c) variation of the residual stress (r22) and the triaxiality with the distance in the crack growth direction respectively (compressive residual stress).

Fig. 20. Comparisons of the computed and measured load–displacement curves for C(T) specimen without residual stress and with compressive residual stress.

the fracture resistance and lowers the specimen’s load carrying capacity. After crack grows away from the residual stress inﬂuence area, the features of crack growth become similar to those exhibited by the virgin specimen. 5.2.2. Compressive residual stress When side-compression is applied behind the crack tip, crack will close and the crack surfaces will contact each other. In ﬁnite element analysis, to prevent crack surface penetration, a rigid surface is added to the symmetric plane behind the crack tip. The ﬁnite element analysis results show that the high compressive residual stress region is at the initial EDM notch (behind the fatigue pre-crack front) and the residual stress distribution is not as uniform as tensile residual stress case, Fig. 19a. Fig. 19b and c shows the variations of the residual stress (r22) and the triaxiality with the distance in the crack growth direction (the crack tip is at x = 0) at the mid-plane respectively.

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Compressive residual stresses reduce the constraint level at the crack front region and tend to close the crack. Consequently, the compressive residual stress increases the fracture resistance. Fig. 20 compares the load–displacement curves of the specimen with compressive residual stress ﬁeld generated by 220 kN side compression with the as received specimen. Included in the ﬁgure are also comparisons between the model predictions and the experimental measurements. Again, good agreement is observed. 6. Concluding remarks In this work, the plasticity and ductile fracture behaviors of an aluminum alloy 5083-H116 are studied through a series of experiments and ﬁnite element analyses. This material’s plasticity behavior is shown to be J3 dependent. A recently developed I1–J2–J3 dependent plasticity model is calibrated and validated by a series of experimental tests and numerical analyses. A ductile failure criterion based on the damage parameter deﬁned in terms of the accumulative plastic strain as a function of the stress triaxiality and the Lode angle is established and the detailed model calibration process is described. With the calibrated model parameters, the 3D fracture locus is plotted in the space of stress triaxiality and Lode parameter. For this material, increasing stress triaxiality greatly reduces the failure strain while change in Lode parameter does affect the failure strain but the effect is not as strong. The plasticity and ductile fracture models are shown to have the ability to predict the crack growth in C(T) specimens with and without side-grooves. In the second part of this investigation, a local out-of-plane compression approach is used to generate residual stress ﬁelds in C(T) specimens and these residual stress ﬁelds are quantiﬁed by ﬁnite element modeling of the side compression process. Tensile residual stress not only increases the crack driving force but also raises the constraint level in crack tip region, which results in lower fracture resistance. Compressive residual stress has the opposite effect. The numerical results, such as load–displacement curves and crack front proﬁles, are compared with experimental measurements and good agreements are observed. The presented fracture model can be further extended to predict the fracture behavior of welded structures where the residual stress ﬁeld can be included in the analysis using the eigenstrain method. Acknowledgements This research is supported by the Ship Structures Committee and the Naval Surface Warfare Center, Carderock Division. References [1] Spitzig WA, Sober RJ, Richmond O. Pressure dependence of yielding and associated volume expansion in tempered martensite. Acta Metall 1975;23:885–93. [2] Brünig M. 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