Numerical simulation of fracture mode transition in ductile plates

Transcription

1 Numerical simulation of fracture mode transition in ductile lates The MIT Faculty has made this article oenly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Xue, Liang, and Tomasz Wierzbicki. Numerical Simulation of Fracture Mode Transition in Ductile Plates. International Journal of Solids and Structures 46, no. 6 (March 29): htt://dx.doi.org/.6/j.ijsolstr Elsevier Version Final ublished version Accessed Thu Dec 27 7:2:45 EST 28 Citable Link Terms of Use Detailed Terms htt://hdl.handle.net/72./96332 Article is made available in accordance with the ublisher's olicy and may be subject to US coyright law. Please refer to the ublisher's site for terms of use.

2 International Journal of Solids and Structures 46 (29) Contents lists available at ScienceDirect International Journal of Solids and Structures journal homeage: Numerical simulation of fracture mode transition in ductile lates Liang Xue *, Tomasz Wierzbicki Deartment of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue 5-28, Cambridge, MA 239, USA article info abstract Article history: Received 22 May 28 Received in revised form 24 October 28 Available online 2 November 28 Keywords: Ductile fracture Damage lasticity Comact tension Fracture mode transition Slant crack Fracture mode of ductile solids can vary deending on the history of stress state the material exerienced. For examle, ductile lates under remote in-lane loading are often found to ruture in mode I or mixed mode I/III. The distinct crack atterns are observed in many different metals and alloys, but until now the underlying hysical rinciles, though highly debated, remain unresolved. Here we show that the existing theories are not caable of caturing the mixed mode I/III due to a missing ingredient in the constitutive equations. We introduce an azimuthal deendent fracture enveloe and illustrate that two cometing fracture mechanisms, governed by the ressure and the Lode angle of the stress tensor, resectively, exist ahead of the crack ti. Using the continuum damage lasticity model, we demonstrate that the distinctive features of the two crack roagation modes in ductile lates can be reroduced using three dimensional finite element simulations. The magnitude of the tunneling effect and the aarent crack growth resistance are calculated and agree with exerimental observations. The finite element mesh size deendences of the fracture mode and the aarent crack growth resistance are also investigated. Published by Elsevier Ltd.. Introduction A thorough hysical understanding of ductile crack initiation and roagation is of essential interest to many scientific discilines and engineering alications. Existing constitutive models are so far not caable of redicting and exlaining several key features of ductile fracture known to exerimentalists, such as the to slant transition of fracture modes in ductile anels. Mathematical simlification has been given to macrocracks in solids such that a singularity is laced at the crack ti in the scoe of conventional fracture mechanics. The treatment for crack advance is described by field variables remote to the crack ti such that the singularity no longer oses roblems. This treatment works for brittle and quasi-brittle materials where the fracture rocess zone is small comared with the secimen geometries and the secimen dimensions are large enough to assess field variables away from the crack ti. In reality, however, these assumtions do not hold for the many metallic materials where large lastic deformation recedes the occurrence of fracture. Crack blunting and necking greatly reduce the accuracy of these simlifications and idealizations. Moreover, the geometry of the solid bodies is often too comlex to calculate and the cracked body may not be idealized as lane strain condition, such that theoretical solutions exist * Corresonding author. Present address: Deartment of Mechanical Engineering, Northwestern University, 245 Sheridan Road, M-, Evanston, IL 628, USA. Tel.: address: xue@alum.mit.edu (L. Xue). (Hutchinson, 968; Rice and Rosengren, 968), due to the constraint length scale in ractical roblems. These issues cannot be resolved without considering the details about the crack rocess zone ahead of a ductile crack. Traditionally, the lane strain models near the crack ti is used in finite element analyses using exlicitly modeled voids in the crack ath (e.g. Gao et al., 26; Kim et al., 27; Xia et al., 995). We show here that, with an additional dimension in the continuum damage lasticity theory, exerimentally observed ductile crack atterns and trends in the crack growth resistance can be redicted using three dimensional finite element simulations. In a laboratory setu, re-cracked lates are often used to study ductile crack roagation. Although some materials show a continuation of a mode I crack (which is normal to the late surface and to the remotely alied load), a transition from a re-crack to a slant mixed mode I/III crack (which is aroximately 45 to the surface) in thin lates is commonly observed in exeriments for many olycrystalline metals and alloys (Anderson, 25; Barsom and Rolfe, 999; Broek, 982; Knott, 973). Fractograhically, a mode I fracture surface is of a fibrous nature and a slant mixed mode I/III fracture surface is a shear tye of failure and is less voided comared with a crack surface (Barsom and Rolfe, 999; Benzerga et al., 24; Cottrell, 965; Pineau and Pardoen, 27; Thomason, 99). However, why and how this transition of the global fracture mode occurs is not fully understood. The vast exerimental slant fracture results (e.g. Dawicke and Sutton, 994; Irwin et al., 958; Mahmoud and Lease, 23; Newman, 985; Srawley and Brown, 965) are in shar contrast with the absence /$ - see front matter Published by Elsevier Ltd. doi:.6/j.ijsolstr.28..9

3 424 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) of slant crack redicted by three dimensional finite element analyses using existing constitutive theories (e.g. Roy and Dodds, 2; Dawicke et al., 995; Gullerud et al., 999; James and Newman, 23; Lan et al., 26; Li et al., 22; Mahmoud and Lease, 24; Newman et al., 23; Tvergaard and Needleman, 26). This discreancy between theories and exeriments is resolved here through a new isotroic continuum damage lasticity theory, which adots a scalar measurement of damage from a three dimensional descrition. We begin with the exerimentally observed differences between the and slant modes in cracked lates. The slant region of fracture is not a mode III crack but involves some ulling aart (mode I). Besides the aarent fracture angle to surface, there are several generally erceived distinctive features and trends between the two aforementioned ductile crack modes, as shown in Fig.. These features and trends include: (i) A crack is usually observed for strong strain hardenable materials; and a slant crack is usually observed for low strain hardenable materials. (ii) Significant necking often recedes the aearance of a ductile crack; while there is often very little neck ahead of a slant crack ti. (iii) A crack front shows significant tunneling in the mid lane; while a slant crack front shows little tunneling throughout the thickness. (iv) Flat cracks are usually found for thicker lates (although shear lis may exist near surface); and slant cracks are more often found for thinner lates. (v) Materials are found to exhibit cracks at a quasi-static loading rate may change to a slant crack uon dynamic loading (see Rivalin et al., 2). These features will be relicated here by a series of numerical simulations. Here we show how mode transition is controlled by the interaction of the ressure and the azimuthal angle of the stress states which dominate the damage accumulation in the lastic rocess zone ahead of the crack ti by extensive finite element simulations. The azimuthal deendence on an octahedral lane for ductile fracture is found to be a missing ingredient that is resonsible for the fracture mode transition. We demonstrate using comact tension secimen that mode transition is determined by the combination of strain hardening caability, the azimuthal deendence and the ressure sensitivity of the fracture characteristics of the material. In the crack rocess zone, the mode of crack is determined by two cometing mechanisms: (i) a attern dominated by the ressure effect with tunneling that can be catured by existing damage theories; (ii) a slant crack about 45 to the surface driven by the azimuthal (the Lode angle) deendence of ductile fracture, which de facto is the controlling factor of mixed mode I/III for thin lates. This Lode angle deendence introduces the effect of the third invariant J 3 of the stress tensor. Our results demonstrate that the azimuthal deendence of ductile fracture together with the ressure sensitivity lays a vital role in determining the fracture attern. We anticiate the resent study to lead to more accurate modeling of ductile fracture at a continuum length scale. For examle, crack redictions in metal forming and failure analyses of large scale structures. 2. Continuum theory of damage lasticity Stried to its essentials, the continuum damage lasticity model consists of () a classical strain hardening and associated flow rule for the lasticity of the matrix material; (2) an evolution law for the ductile damage to deict the microstructural rearrangement along the lastic loading ath; and (3) a damage couled yield condition to account for the material deterioration due to the microstructural change. For simlicity, we emloy von Mises yield criterion for the matrix which only deends on the second invariant of the stress deviator. The evolution law of ductile damage described by a three dimensional cylindrical decomosition (Xue, 27a), which incororates all three stress invariants, i.e. I ; J 2 and J 3. The evolution of damage resembles the evolution of equivalent stress in conventional lasticity theories. Here, we summarize this damage model below. Because the lastic damage is ath deendent, the damage is given in the rate form, which relies on the concet of fracture enveloe. A fracture enveloe is defined in the three dimensional sace of the triaxial lastic strain lane and the hydrostatic ressure. The fracture enveloe is characterized by the lastic strains at which material fracture occurs from all ossible loading aths of constant ressure and constant azimuthal angle. The ressure sensitivity and the azimuthal deendence of the fracture strains are described by a ressure deendence function l ðþ and an azimuthal deendence function l h ðh L Þ, resectively. An illustrative fracture enveloe for a ductile material is sketched in Fig. 2. The vertical axis is the mean stress (i.e. negative ressure) and the triaxial horizontal lane is the rincial lastic strain lane, which characterizes the azimuthal angle on an octahedral lane when the lastic deformation is assumed to be isochoric. Many materials, such as rocks and metals, exhibit higher ductility under high comressive ressure. This henomenon has been extensively studied in the ast century (e.g. see monograhs by Bridgman (952) and Pugh (97)). In the resent model, the azimuthal deendence of the fracture enveloe is described by six eaks (at generalized tension and comression conditions) and six valleys (at generalized shear conditions). In other words, ductile fracture of solids is more sensitive to shear tye of loading. This consideration is based on the observation of many exeriment results (see e.g. Bao and Wierzbicki, 24; Barsoum and Faleskog, 27; Clausing, 97; McClintock, 97; Wilkins et al., 98). It should be noted that convexity does not aly to fracture enveloe, a b Fig.. The two distinct modes of crack in ductile lates (a) mode; (b) slant mode. Shaded areas are the cross-sections erendicular to the crack roagation direction. Red lines indicate the crack fronts. (For interretation of the references to color in this figure legend, the reader is referred to the web version of this aer.)

4 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) l h ðh L Þ¼cþð cþ 6jh k Lj ; ð6þ where q, lim, c and k are material constants, is the current ressure and h L is the Lode angle ðh L 2½ =6; =6ŠÞ. The Lode angle is one of several arameters that are commonly used to denote the azimuthal angle on an octahedral lane in the rincial stress sace. The Lode angle is defined by the rincial stress comonents h L ¼ tan ffiffiffi 3 h L ¼ 3 sin r 2 r r 3 r r 3 J 3 r 3 eq! or ; ð7þ Fig. 2. A three-dimensional reresentation of the fracture enveloe in the sace of the rincial lastic strains ðe ; e 2 and e 3Þ and the mean stress r m. since it is a collection fracture strains from rescribed loading aths and is based on measurement of strains rather than stresses. For isotroic materials, the azimuthal angle can be characterized by the Lode angle, which varies from /6 to /6 and is defined for each sextant of the octahedral lane. Here, we restrict ourselves to isotroic materials such that the fracture enveloe is eriodic, i.e. it is identical in each of these six sextants. We start by searating the matrix material, which is defect free, from the macroscoic structure of solid. The weakening effect due to the deterioration of the material is introduced by the damagecouled yield condition: U ¼ r eq wðdþr M 6 : ðþ For arbitrary lastic loading ath, the damage is calculated by the following integral: Z ec ðm Þ de D ¼ 6 ; ð2þ e f m e e f where m is a material arameter, e f is a fracture strain enveloe defined on the stress state and e c is the critical strain at which fracture occurs. It is assumed D ¼ for virgin material and D ¼ for a comlete loss of load carrying caacity, i.e. fracture occurs. It can be verified that for loading aths with constant fracture strain e f the above definite integration reaches unity when e c ¼ e f. We assume the Young s modulus of the material decrease as damage accumulates, i.e. EðDÞ ¼wðDÞE, where E is the original undamaged Young s modulus. The weakening function wðdþ in Eq. () is described by wðdþ ¼ D b ; where b is a material constant to be calibrated from fitting exerimental curves. The fracture strain enveloe is defined on the current ressure and the Lode angle h L only e f ð; h L Þ¼e f l ðþl h ðh L Þ; where e f is a material constant and l ðþ and are l h ðh L Þ the ressure sensitivity function and the azimuthal angle deendence function, which in the resent study adot a logarithmic form of ressure deendence function: ( l ðþ ¼ q log lim ; P lim ½ exð=qþš; ð5þ ; < lim ½ exð=qþš; and the second kind of Lode angle deendence function: ð3þ ð4þ where r ; r 2 and r 3 are the ordered rincial stress comonents and J 3 ¼ s s 2 s 3 is the third stress invariant where s ; s 2 and s 3 are the ordered rincial stress deviator comonents. In this set of constitutive equations, six material arameters are used in total. These material arameters are a reference strain e f, two for ressure deendence function lim, q, two for the Lode angle deendence function c and k, one for the damage accumulation exonent m and one for the weakening effect b. These arameters are treated as constants for a given material and are to be calibrated from exeriments. Xue and Wierzbicki (submitted for ublication) resented a combined exerimental and numerical rocedure to calibrate aluminum alloy 224-T35 using a series of tests at different mean stresses and Lode angles. To summarize the above method, a nonlinear integral for damage is adoted (Eq. (2)). In Eq. (2), the integrand, i.e. the accumulation rate of damage (a non-negative value), is imlicitly influenced by the ressure and the Lode angle of the current stress state by the resective effects on the fracture enveloe e f for a given incremental lastic strain _e. The weakening effect of the accumulated damage enters the yield condition through a weakening factor wðdþ, which deends on the magnitude of damage at the current state. The strain hardening effect enters the yield condition through the matrix stress strain relationshi r M, which is a function of the equivalent lastic strain. The stress integration rocedure for exlicit algorithm is summarized in Aendix A. In a revious aer, Xue and Wierzbicki (28) adoted the resent theory and found that the synergistic combination of the Lode angle deendence and the weakening factor governs a slant mode in comact tension secimens and romotes shear lis. In the resent aer, we further exlore the transition of fracture mode on various material and geometry asects of the comact tension tests. These factors include the Lode angle sensitivity arameter, the strain hardening, the ressure sensitivity and the thickness of the comact tension late. Their effects on the fracture mode, the tunneling of crack front and the aarent crack growth resistance, R, are discussed. 3. Modeling We consider a base scenario of a comact tension secimen according to ASME E399 with width W = 5.8 mm, thickness B = 6.35 mm, a re-crack of 6 notch angle (normal to surface) and a crack to width ratio a=w ¼ :5. The external load is alied in the two cylindrical holes through ulling aart of two frictionless ins inside the holes in a single stroke. No fatigue crack is considered. In Sections 4 6, the comact tension secimen is discretized into 45,4 8-node reduced integration elements. Twenty elements are used in the thickness direction and the asect ratio of the elements along the crack ath is about ::. The element size in central zone where the crack ath is anticiated is aroximately.3 mm.3 mm.3 mm (W L H). Same mesh size are used in Section 7 for the thickness deendence simulations,

5 426 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) where additional elements are added in the thickness direction. In Section 8, the base scenario comact tension secimen is adoted for the mesh size deendence simulations, where element size is varied. All simulations are carried out using LS-DYNA with the damage lasticity model imlemented as a user subroutine. The stress integration rocedure is included in Aendix A. The matrix stress strain relationshi is assumed to follow the Swift tye of ower law relationshi r M ¼ r y ð þ e =e Þ n ; where r y is the initial yield stress of the matrix, n is the hardening exonent, e is a reference strain, e is the lastic strain and r M is the equivalent matrix stress. In the resent study, we focus on the deendence of fracture modes and crack growth resistance on the strain hardening, the Lode angle deendence and ressure sensitivity. In order to draw conclusions from these influencing factors, we adot the following set of fixed arameters in the numerical study Young s modulus E = 7 GPa, Poisson s ration m =.3, mass density q ¼ 27 kg=m 3, r y ¼ 3 MPa, e ¼ :8, e f ¼ :8, lim ¼ MPa, k ¼ :, m ¼ 2:, b ¼ 2: and n, q and c will be varied to erform a arameteric study. The choice of fixed arameters is based on an aluminum alloy tested and calibrated for the damage lasticity model (Xue, 27b; Xue and Wierzbicki, submitted for ublication). 4. Effect of azimuthal deendence In this section, studied is the deendence of the crack roagation atterns on the material arameters of c. Material constants n =.2, q =. are chosen for this examle. The mixed mode I/III and mode I of cracks are shown in Fig. 3. Fig. 3(a) shows a transition from a starting crack with tunneling to a slant crack throughout the thickness direction for c =.5. Fig. 3(b) shows continuing crack roagation in the entire thickness for c =.7. The difference in the azimuthal sensitivity of the fracture enveloe triggers the change in the crack mode. With diminishing azimuthal deendence of ductile fracture ðc! :Þ, the comact tension crack changes from a slant crack to a one. This is consistent with the result in Xue and Wierzbicki (28) that shows a slant crack can only be redicted for materials with both effects of the azimuthal angle deendence and the weakening of material strength. Fig. 4 shows a comarison of the two different modes while roagating along the ligament. Note the differences in the crack fronts, where roagating crack shows a more significant tunnel in the mid section. A fully develoed slant crack is relatively straight comared with a arabolic curved crack front. The extent of necking in the ath of the stable crack roagation is indicated by the distance from the fracture edge to the edges of ð8þ the undeformed late which are indicated by the horizontal thin lines in Fig. 4. In these simulation results, the shrinkage in the thickness direction at the vicinity of crack is more severe for the mode I case than the mixed mode I/III case. This agrees with feature (ii) of exerimental observations. The load versus load line dislacement and the normalized tunneling versus mid-lane crack extension for various c values are lotted in Fig. 5(a) and (b). For a straight re-cracked late, the fracture starts at the mid-lane and roagates both forward and laterally to the surface. The crack front is usually a arabolic shae, and is thus often called tunneling. The extent of tunneling is defined as the difference of crack extension at mid-lane and at surface divided by the original late thickness (Dawicke and Sutton, 994). For mode I, the tunneling remains at a high level after initial crack forms over the entire thickness. However, for slant crack roagation, the tunneling dros sharly after a maximum value is reached. This distinct feature of tunneling is identified in Fig. 5(b) and (c) as the shae of tunneling evolution curves can be categorized into two grous deending on the c values. The simulation results agree with exerimentally observed feature (iii). The mode transition from slant to mode occurs at about c =.65 in this case. There exists a stee transition of average tunneling between the two full fledged roagation modes (as shown in Fig. 5(c)). On the left-hand-side of the mode transition line, slant cracks develo and the stable normalized tunneling increases with increasing c. On the right-hand-side of the mode transition line, cracks aear and the stable normalized tunneling decreases with increasing c, which indicates the mid-lane crack extension is increasingly longer than the crack extension at the surface for decreasing c values. The advance of mid-lane crack is understood as the mid-lane material is subjected to more severe lane strain condition than the surface materials; therefore, the mid-lane material is more rone to fracture for lower c values. However, such advances in the mid-lane are not sustainable as the material arameter c dros further. A global mode transition occurs when the crack finds itself an easy ath to roagate divergently to surfaces at an aroximately 45 angle. Consequently, shear lis form at the tail of the crack near surfaces. The shear lis grow and eventually merge to form a slant crack over the entire thickness. When oosite shear lis form, a small ortion of anti-symmetric load is introduces to the nominally symmetric loading system. This can be seen in Fig. 4 as the remote edges of a slant cracked secimen (horizontal thin lines) are no longer overlaing, which further romotes a mixed mode I/III to form. For a cracked secimen, the loading system remains symmetric with resect to the midlane; therefore, the remote edges remain overlaing (Fig. 4(a)). It is also noted that the aarent crack growth resistance is found to be a strong function of the material arameter Fig. 3. Crack atterns of comact tension secimens. (a) An overall slant crack ðc ¼ :5Þ and (b) a crack ðc ¼ :7Þ roagate in the ligaments of comact tension secimens under isothermal condition. (c) Same secimen for ðc ¼ :7Þ changes to a slant mode under adiabatic loading.

6 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) Fig. 4. A comarison of crack front evolution: (a) a crack becomes a slant crack and (b) a continued crack. 4 7 a γ =. b c d γ mode transition line = γ=.3 6 γ =.4 γ =.5 γ =.6 γ = γ =.8 8 γ =.9 γ= Load line force (kn) Normalized tunneling.2. Average normalized tunneling.2. slant Resistance R (kjm 2 ) 3 2 mode transition line slant 5 Load line searation (mm) 5 5 Crack extension (mm).5 γ.5 γ Fig. 5. The load-dislacement curves for different c values (a) and tunnelling (b) and (c). A transition of the tunneling occurs when the global fracture mode changes (b) and (c). However, such mode change has little effect on the trend of the aarent crack growth resistance (d). (Fig. 5(d)). The aarent crack growth resistance is calculated for the steady crack roagation by integration the load dislacement curve with resect to load line dislacement and divided by the cracked ligament area in the original configuration. It is a characteristic of the energy dissiated er unit area of fracture of the material under stable crack advancing. The aarent fracture area is obtained by counting the fractured elements in the rojection lane erendicular ffiffiffi to the load line. (NOT the fracture surface area, which is about 2 times of the aarent fracture area for slant crack.) It is shown in Fig. 5(d) that the aarent crack growth resistance grows steadily and no obvious transition in the sloe is found at the mode transition line. Considering the crack advancing, we focus on the damaging and fracturing sequence at the crack ti rocess zone. A close examination of the stress state reveals two cometing damaging mechanisms ahead of the crack ti driven by the mean stress effect and the azimuthal angle effect, resectively. In the three-dimensional fracture enveloe shown in Fig. 2, these two mechanisms can be grahically interreted as () the mean stress effect ushing uwards and the fracture enveloe shrinks; and (2) the azimuthal angle effect dragging into the valleys (where the Lode angle h L = ). In both cases, the damaging rocess is accelerated at constant lastic strain rate. The stress state histories of the fractured elements at three different locations () at surface, (2) at /4 thickness and (3) at midlane are lotted in Fig. 6. Several trends emerge from the stress state histories at different locations along crack aths. Firstly, the azimuthal sensitive factor l h has more contribution in a slant crack, while the ressure sensitive factor l is more imortant in a crack (see the relative osition of lines marked by square and triangle in the right sublots, Fig. 6(d) and (h)). In both cases, ressure effect has more influences near the center of the late due to the buildu of constraint in the thickness direction. While the damage-averaged stress state sensitive factors l h and l do not vary much in the thickness direction in our calculation, it is noted that the ressure sensitive factor for the slant crack increases sharly near the surface for a slant crack (Fig. 6(d) and (h)). Note a larger sensitivity factor means less influential to damage accumulation. Comaring with a crack, the mean stress factor does not change much at the center of the late for the slant case. The relatively straight crack front and very little necking in the slant mode reduce the constraint in the thickness direction near the surface (see the line marked by square in Fig. 6(d)). Secondly, the ressure factors ðl Þ increase when the materials aroach their final fracture oints, which indicate decreasing damage rate. This means the mean stresses decrease before onset of fracture. The reason for this is that the mean stresses reduce as the Young s moduli decrease and, therefore, the initially high mean stresses cannot be hold. In a way, the revious elastic volumetric strain is now generating less mean stress due to a weakened bulk modulus. Finally, another noticeable feature in the time history of the fractured elements is that there exists a major eak in the azimuthal deendence factor (dash dot curves in Fig. 6(a) (c) and (e) (h)) before the element is fractured and removed. This eak is found for all three locations in the thickness direction for both

7 428 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) a SLANT CRACK at surface b c at /4 thickness at mid lane d μ μ.2 μ.2.2. Damage and stress sensitivity factors μ * D Plastic strain ε Damage and stress sensitivity factors μ * D Plastic strain ε Damage and stress sensitivity factors μ * D Plastic strain ε Average stress state sensitivity factors μ μ *.5 Thickness osition Damage and stress sensitivity factors FLAT CRACK e f g h at surface μ.2 μ * D Plastic strain ε Damage and stress sensitivity factors at /4 thickness μ.2 μ * D Plastic strain ε Damage and stress sensitivity factors at mid lane μ.2 μ * D Plastic strain ε Average stress state sensitivity factors μ μ *.5 Thickness osition Fig. 6. Time histories of damage ðdþ, ressure sensitivity factor ðl Þ, azimuthal sensitivity factor ðl h Þ and their roduct at three different locations (a c; e g) in the thickness direction for the two fracture modes. The average values of the stress state sensitive factors with resect to damage (d and h) indicate that the governing factors for and slant fracture modes are driven by the ressure sensitivity and the Lode angle deendence, resectively. and slant crack modes. An examination in the stress history of the fractured elements shows that this eak is due to the quick buildu of the stress comonent in the thickness direction (Z), which surasses the stress comonent in the crack roagation direction (X), while the stress comonent in the loading direction (Y) remains the maximum in the fracture rocess zone. Therefore, the Lode angle of the stress state turns from a close-to-lane-strain condition (a valley in the fracture enveloe) to a generalized tension condition (a eak in a fracture enveloe) and then again goes to the other side of the eak. Thus, a eak is created in the history of the azimuthal deendence factor. The formation of the shear slis can be illustrated by taking a closer look at the contours of the two cometing stress state deendent factors at a cross-section of the anel as shown in Fig. 7. The cross-sections are similar to that of the cu-cone fracture of a round bar but the comact tension secimens are more constrained in the roagation direction due to the length of ligament. For a high c value (c =.7 as in Fig. 7(a) (d)), the crack roagation direction is governed by the ressure sensitivity where does not vary much in all directions ahead of the crack ti and dictates the crack. On the contrary, for a low c value (c =.5 as in Fig. 7(e) (h)), the crack roagation direction is dictated by the azimuthal deendence factor which forms fast damaging zones at about 45 to the surface (dark areas) and overshadows the ressure deendence factor as shown in Fig. 7(f) and (g). 5. Effect of adiabatic heating In the dynamic resonses of materials, adiabatic heating induces thermal weakening that can trigger shear localization in the heated zone (Bai and Dodd, 992; Wright and Batra, 985). The formation of adiabatic shear bands is critical in certain imact and enetration roblems. Rivalin et al. (2) conducted both quasi-static and dynamic exeriments on re-cracked steel lates. In their exeriments, a transition from quasi-static crack to a slant dynamic crack was found, which suggests the adiabatic behaviour of material can trigger mode transition in dynamic crack roagation. Mathur et al. (996) analyzed a very thin re-cracked late (thickness =.33 mm) and showed adiabatic shear banding forms at the sli lanes aroximately 45 to the surface in a voided media. For adiabatic loading condition, a classic lasticity work dissiation induced thermal effect is introduced. The yield stress is not generally sensitive to strain rate for aluminum alloys (Zhang and Ravi-Chandar, 26) and is therefore neglected in the resent study. Under the adiabatic condition, the local temerature increase is calculated through the lastic work dissiation, i.e. ot qc ot ¼ ar : d ; ð9þ where q is the material density, C is the heat caacity, t denotes time, T denotes temerature, r is the stress tensor, d is the lastic

8 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) Fig. 7. Contour lots of the ressure sensitivity factor ðl Þ, azimuthal sensitivity factor ðl h Þ and their roduct at a artially cracked cross-section for and slant fracture modes. The final through thickness cracks are shown in the right sublots. The azimuthal sensitive factor dictates a slant crack, while the ressure sensitive factor dictates in a crack. A smaller sensitivity factor (indicated by dark area) means more influential to the damage accumulation. Panels (a) (d) are for c =.7 case and anels (e) (h) are for c =.5 case. art of rate of deformation tensor and a is Taylor Quinney coefficient, which is taken to be.9. A modified Johnson Cook thermal weakening factor to the material strength is adoted (Zhang and Ravi-Chandar, 26). The temerature deendent matrix strength is characterized by the two distinct fracture modes. The results are resented in Fig. 8. The abscissa denotes the azimuthal deendence arameter c where on the left end c =. means ideal nil ductility in general- r M ¼ r y ð þ e =e Þ n ½ðT T ref Þ=ðT melt T ref ÞŠ m T ; ðþ where T is the current temerature of the material, T ref ¼ 297 K is a reference temerature (e.g. room temerature where the exeriments are conducted), T melt ¼ 755 K is the melting temerature for aluminum alloy and the exonent m T is a material constant, which is chosen to be. in the resent study. With the thermal weakening, the crack under adiabatic condition turns into a slant crack for n ¼ :2 and c ¼ :7 material (Fig. 3(c)), which should be comared with a crack for the same material under isothermal condition (Fig. 3(b)). This demonstrates the exerimentally observed feature (v). 6. Effect of strain hardening Exerimental results suggest that materials with strong strain hardening caability are more likely to fracture in a attern (Newman, 985; Pardoen et al., 24; Pineau and Pardoen, 27). To further investigate the hardening effect, a series of numerical simulations using the same finite element model is carried out for varying strain hardening exonent n and azimuthal deendence arameter c to determine the searating line between Fig. 8. A fracture mode transition line is determined from a series of numerical simulation results where the strain hardening exonent n and the Lode angle deendence arameter c are two varying material constants. Triangles denote slant cracks; squares denote cracks.

9 43 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) a n=.5 b c n=. mode transition line 45 d 6 n=.5 n= n=.25 4 n=.3 n= n=.4 n=.45 n= n= slant Load line force (kn) 4 2 Normalized tunneling.2. Average normalized tunneling.2. Resistance R (kjm 2 ) 5 mode transition line f lat slant 5 Load line searation (mm) 5 5 Crack extension (mm) Hardening exonent n Hardening exonent n Fig. 9. Fracture characteristics varies with resect to the strain hardening exonent n. Note the distinct normalized tunneling for slant and fracture modes for different hardening exonent n. The aarent crack growth resistance increases for small n but decreases for large n. ized shear condition and on the right end c =. means the fracture enveloe is indeendent of the Lode angle. The coordinate is the strain hardening exonent n which covers a wide range of values for common metals and alloys. In Fig. 8, near the limiting cases of the transitional boundary, for n ¼ :5; c ¼ :45 (circle), the crack turned to slant crack after roagates about mm. For n ¼ :5; c ¼ : (diamond), two tiny shear lis form near free surface and leave a zone in the middle; the shear lis never join into a through thickness slant crack. The results shown in Fig. 8 that the material arameters lead to slant fracture attern locate at the low left corner of the c n matrix (indicated by a light green color ). The remaining regime where a crack is found is shown in grey color. A transitional boundary is also shown in Fig. 8 as a solid line. Above the boundary line, the strain hardening exonent is large and the mean stress influence is significant in the anel, which leads to a crack. This agrees the well-known exerimental observations, i.e. exerimentally observed feature (i). Another attern emerges from Fig. 8 is that low Lode angle deendence (i.e. larger c values) tends to result in a crack. This can be seen from the right-hand side of the boundary, the Lode angle deendence of fracture is diminishing and a crack is observed. However, this effect is not known reviously due to the inadequate awareness of the azimuthal deendence of ductile fracture in the literature. Further exerimental verification is in need in this regard. Similar to Fig. 5, we draw conclusions from Figs. 9 and on the deendence of fracture characteristics with resect to the strain hardening exonent n (for fixed c =.5 and q =.) and the ressure sensitivity arameter q (for fixed c =.5 and n =.2). In Figs. 9 and, slant cracks are found in the intermediate range of n and q values. For small n, the deformation is highly localized in the crack lane. When constraints are build-u in the thickness direction, significant tunneling roagates in the mid-lane (Fig. 9(c)), which is followed by lateral roagation instead of forming shear lis at the tail of the crack. For large n, excessive ressure build-u drives the material to fracture in a attern. It should be noted that the fracture transition between intermediate and large values of n, the crack growth resistance eaked and then dros when n > :25 (Fig. 9(d)). For interretation of the references to color in Fig. 8, the reader is referred to the web version of this aer. For small q, because of the diminishing ressure sensitivity, excessive necks form ahead of the crack ti and result in an overall crack in the significantly reduced section thickness due to geometrical influence. In this case, the influence of geometrical change in the fracture rocess zone is significant. The lastic strain rate (in Eq. (2)) in the normal section is much greater than the rest. For large q, the material is highly sensitive to ressure and breaks in cleavage manner. This can be seen from Fig. 9(d) that the crack growth resistance dros to almost zero for q > 2:. From the vertical line of c = in Fig. 5, without the consideration of the Lode angle deendence of ductile materials, the redicted fracture modes are all cracks. Because of this reason, all numerical studies ublished in the literature have not been able to redict a slant crack and to cature the five distinctive features listed in Section. As for the formation of shear lis, Tvergaard and Needleman (984) simulated the -to-slant transition in the cucone ruture of round bars using micromechanical modified Gurson model (Gurson, 977) and exlained the formation of shear lis near surface due to void sheeting mechanism of secondary articles. The cu-cone transition can also be found in numerical simulations using continuum damage mechanics (Teng, 28 etc.). However, the modified Gurson model and the conventional continuum damage mechanics method do not include the effect of third stress invariant (in other word, c =. is tacitly assumed). From Fig. 8, under the resent assumtions, a slant crack in thin late cannot be redicted without introducing additional effects, such as adiabatic heating (Mathur et al., 996) or anisotroy (Besson et al., 2). Further modifications to the Gurson-tye models are also introduced by Xue (26) to include damages associated with void shearing that is deendent on the Lode angle (Xue, 27b, 28). Similar remedy was adoted by Nahshon and Hutchinson (28) to include a third stress invariant deendent damage evolution law more recently. 7. Effect of late thickness The general oinion about crack growth resistance for lates is that it increases at low thickness where a slant crack is observed and decreases with resect to thickness as mode I crack becomes dominant. Eventually, the crack growth resistance reaches its asymtotic value of the mode I fracture for very thick late. This henomenon is well known to the exerimentalists when measuring the fracture toughness for ductile materials using lates of

10 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) a q=. b c d q=.25 mode transition line.9.9 q= q=.375 q= q=. 6 q= q=2. q= slant Load line force (kn) Normalized tunneling Average normalized tunneling Resistance R (kjm 2 ) mode transition line slant 5 Load line searation (mm) 5 5 Crack extension (mm) 2 3 q 2 3 q Fig.. Fracture characteristic varies with resect to the ressure sensitivity arameter q. The aarent crack growth resistance decreases with increasing ressure sensitivity (increasing q values). The fracture mode does not have an obvious effect on R value (d), but slant crack significantly reduces the tunneling effect (c). different thicknesses (Broek, 982, Chater 4; Barsom and Rolfe, 999, Chater 4; Anderson, 25, Chater 2). A henomenological exlanation is given by Krafft et al. (96) who searated the energy dissiation of the shear lis (slant region) from the grossly central zone in fractured thick lates. They exlained the increase of crack growth resistance in the small thickness regime is due to the energy dissiation in a quadratic form to the deth of shear li and in the large thickness regime the deth of shear lis is bounded and eventually only energy associated with mode I is dissiated. We simulate the fracture of comact tension secimens for various thicknesses and the crack growth resistances are calculated. Simulations are erformed for a series of comact tension lates for n ¼ :2, c =.5 and with thickness various from.5875 to 25.4 mm. The.5875 mm thick late buckles in the ligament and therefore is excluded from the following discussion. The results are shown in Fig.. From Fig., the center ortion of the crack surface becomes as the thickness of the lates increases. For thinner lates (thickness equals 6.35 mm and less), an overall slant mode is observed throughout the thickness. For thicker lates (greater than 6.35 mm), the secimens show () a central zone where materials fail in mode I oening mode and (2) two shear li zones where mixed I/III mode slant cracks form. Due to the differences in fracture mechanisms of the and slant regimes, the aarent crack roagation resistance varies with the late thickness. The aarent crack growth resistance is comuted from the crack surface in the original configuration rojected to the initial crack lane. The trend obtained from the numerical simulation reresents the well-known shae of thickness deendence curve of the crack growth resistance, as shown in Fig. 2(d). The maximum thickness calculated here is 25.4 mm, which is limited to the excessive comutational time. It aears that the lane strain limit has not been reached yet for this hyothetical material. It is also noticed that the aarent crack growth resistance R does not decrease immediately after the aearance of a region at the center of the fracture surface. Rather, the aarent crack growth resistance continues to increase until thickness reaches about mm when the region consists of about 2% of the thickness. The ortion of in the thickness direction increases with increasing thickness as shown in Fig. 2. This matches the exerimental observation feature (iv). 8. Mesh size deendence In the late stage of deformation of an element, the hardening modulus of the material becomes non-ositive when damage becomes imortant. The numerical solution is mesh sensitive as this tye of damage constitutive relationshis usually do. One way to regulate the deformation is to include a characteristic length scale in the constitutive model or a strain gradient deendent term in the damaging rocess. The resent damage lasticity model does not incororate such an inherent length scale, therefore, the solu- Fig.. The aearance of crack surface varies with the late thickness. With increasing late thickness, the zone initiates at the center and grows.

11 432 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) a mm b.4 c.7 d 6.35 mm mm mm.2.6. mm 2.7 mm 9.5 mm mm Load line force (kn) 3 2 Normalized tunneling % Flat region Resistance R (kj/m 2 ) Load line searation (mm) 5 5 Mid lane crack extension 2 3 Plate Width (mm) Plate Width (mm) Fig. 2. The load-dislacement curves (a) and the relationshis between the tunneling (b), the ercentage of the ortion of the crack surface (c), the aarent crack growth resistance (d) and the late thickness are lotted for the simulation results. The ortion of the fracture surface increases with increasing late thickness. The aarent crack growth resistance eaked at about mm late thickness. tion is mesh deendent. Moreover, the slant mode of fracture is usually localized in narrow shear bands. The characterization of these shear bands certainly relies on the size of the mesh. Coarse mesh may not be able to deict the details in a shear band and lead to a different fracture mode (in the resent study, a mode is found instead for coarse meshes). A series of numerical simulations were conducted to further investigate the deendence of mode transition on the mesh size. The 6.35-mm-thick comact tension secimen is discretized by different size of elements. We denote the element size by the number of elements in the thickness direction of the late. The number of elements in the thickness direction varies from 5 to 25 elements (element size mm). The asect ratio of the elements along and in the neighborhood of the crack ath is ket ::. The matrix stress strain curve remains the same as the base scenario, where the hardening exonent n =.2. The ressure deendence arameters are the same. We also exlore the deendence of the mode transition on the Lode angle deendence arameter c, which has been identified as one of the key arameters for the mode transition in late. The results with resect to Fig. 3. The numerical results of fracture mode deend on the mesh size and the Lode angle deendence factor c. For examle, the critical number of elements in the thickness direction is 5 for c =.5 to cature a slant mode in this case (element size.423 mm). the element size and the Lode angle deendence are lotted in Fig. 3. From Fig. 3, smaller element size are more caable of caturing a slant mode of fracture for a fixed c. There aears a transition line between the and the slant modes of fracture. The transition line is lotted in Fig. 3 as a thin solid line. The simulations where a slant mode is found are indicated by a triangle and where a mode is found are indicated by a square. The critical number of elements in the thickness direction to cature a slant mode deends on the tendency of the material on how easy a shear band forms. For instance, at low c values (e.g. c 2½:; :2Š), less than elements are need to cature a slant mode (element size.635 mm). At higher values, e.g. c =.5, 5 elements are needed (element size.423 mm). For the number of through thickness element greater than 8, the slant mode of failure ersists for all c values when other material arameters are fixed. Nine load dislacement curves for different mesh sizes of the same comact tension secimen and same material arameters are lotted in Fig. 4. For the same set of material arameters, the load dislacement curve dros when the mesh is finer. The crack attern and energy dissiation when crack roagates aear to be sensitive to the mesh size. At low resolution of discretization (the number of through thickness elements between 5 and 2), the numerical results show a crack, which indicates the shear bands in the thickness direction cannot be catured. The significance of the tunneling effect becomes more obvious when the mesh size decrease (see Fig. 4(b) and (c)). When the number of through thickness elements is equal or greater than 5, the resolution of the mesh becomes caable of deicting the shear bands. In these simulations, shear lis form first and then the entire crack front changes to a slant attern after a short transitional area. Due to the diminishing constraint in the thickness direction for a slant crack, the tunneling effect also decreases when the mesh becomes finer. The aarent crack growth resistance is a monotonic decreasing function with the number of elements in the thickness direction. When elements become smaller, the crack rocessing zone is characterized in more details. The deformation is more localized in the shear bands such that the energy dissiation is narrowed to a smaller zone. This yields the continuous decreasing of the aarent crack growth resistance with finer mesh. It also aears that the mode of crack does not have an obvious imact on the aarent crack resistance. The transition line between the two modes deending on the mesh size is marked in Fig. 4(c) and (d).

12 L. Xue, T. Wierzbicki / International Journal of Solids and Structures 46 (29) a b c d mode transition line 7 mode transition line Load line force (kn) el=5 el=8 el= el=2 el=5 el=8 el=2 el=22 el= Load line searation (mm) Normalized tunneling Mid lane crack extension Average normalized tunneling slant 2 3 Number of Thickness Elem. Resistance R (kjm 2 ) slant 2 3 Number of Thickness Elem. Fig. 4. The load-dislacement curves (a) and the relationshis between the tunneling (b) and (c), the aarent crack growth resistance (d) and the number of through thickness elements are lotted for the simulation results. 9. Conclusions We have determined that the crack mode in ductile anels is governed by the interaction of the ressure sensitivity and the Lode angle deendence characteristics of the material. The Lode angle deendence was a missing ingredient in the constitutive characterization of the ductility of material in existing theories. By including this additional dimension, the underlying cometing fracture mechanisms are revealed. The novelty of this work also resides in the systematic investigation of the influencing factors of the strain hardening, the Lode angle arameter and the hydrostatic ressure arameter on the ductile fracture characteristics that was not fully exlored and addressed in the literature. Exerimentally observed distinct fracture modes are reroduced by a series of finite element simulations. It is remarkable that all major features of the deendence of mode transition uon the material hardening caacity, the Lode angle deendence arameters, the tunneling effect, the adiabatic heating induced mode transition and the thickness deendence of aarent crack growth resistance are catured using the newly develoed continuum theory of damage lasticity. Also investigated in the resent research is the mesh size deendence of ductile fracture. It is well known that the numerical solution of this tye of damage model without an inherent length scale is mesh size deendent when fracture is concerned. In the comact tension case, the shear bands cannot be correctly reresented when the resolution of the mesh is low. A series of simulations for different mesh sizes and the Lode angle deendence arameters c is conducted. A crack is observed for relatively coarse mesh and a slant crack is observed for finer mesh when the geometry and the material constants remain the same. The aarent crack growth resistance is found to decrease with decreasing element size. Further elucidation of the local damaging rocess at the crack ti requires higher resolution in the deendence functions on the mean stress and the azimuthal angle, the damage evolution law and the damage-couled yield function. Given the state-of-art exerimental technique, the determination of these functions remains difficult. We envision a continued rogress in describing and redicting ductile fracture under the roosed framework. Acknowledgment Suort of this work came in art from the ONR/MURI award to MIT through the Office of Naval Research. Aendix A For illustration urose, the evolution of the deviatoric stress tensor is grahically reresented in Fig. 5 for the one-dimensional case, where the deviatoric stress goes from s n at time t n to s nþ at time t nþ with weakening considered. The abscissa is the total strain. An exlicit stress integration rocedure is used in the calculation in the resent study. At time t n, two state variables, e n and D n and the stress state r n are given 2. The stress integration rocedure is the following: Given: fr n ; e n; D n g and De ¼ e nþ e n at time t n. Ste : Calculate e e n ¼ 9 wðd C nþ : r n ; e y ¼ e e n þ De; >= ^e ¼ e y 3 ðtrey Þ; qffiffiffiffiffiffiffiffiffiffiffi ~e ¼ ^e : ^e ; 2 3 >; r trial nþ ¼ r n þ wðd n ÞC :½DeŠ; ðþ ð2þ Fig. 5. A schematic drawing illustrates the evolution of the deviatoric stress tensor for the one-dimensional case. 2 Bold faced letters are tensors.