A Mechanism-Based Approach for Predicting Ductile Fracture of Metallic Alloys

Transcription

1 The University of Akron Mehanial Engineering Faulty Researh Mehanial Engineering Department 6-13 A Mehanism-Based Approah for Prediting Dutile Frature of Metalli Alloys Xiaosheng Gao University of Akron, main ampus, xgao@uakron.edu Please take a moment to share how this work helps you through this survey. Your feedbak will be important as we plan further development of our repository. Follow this and additional works at: Part of the Mehanial Engineering Commons Reommended Citation Gao, Xiaosheng, "A Mehanism-Based Approah for Prediting Dutile Frature of Metalli Alloys" (13). Mehanial Engineering Faulty Researh This Conferene Proeeding is brought to you for free and open aess by Mehanial Engineering Department at IdeaExhange@UAkron, the institutional repository of The University of Akron in Akron, Ohio, USA. It has been aepted for inlusion in Mehanial Engineering Faulty Researh by an authorized administrator of IdeaExhange@UAkron. For more information, please ontat mjon@uakron.edu, uapress@uakron.edu.

2 A Mehanism-Based Approah for Prediting Dutile Frature of Metalli Alloys Xiaosheng Gao Department of Mehanial Engineering, The University of Akron, Akron, OH 4435, USA Abstrat Dutile frature in metalli alloys often follows a multi-step failure proess involving void nuleation, growth and oalesene. Beause of the differene in orders of magnitude between the size of the finite element needed to resolve the mirosopi details and the size of the engineering strutures, homogenized material models, whih exhibits strain softening, are often used to simulate the rak propagation proess. Various forms of porous plastiity models have been developed for this purpose. Calibration of these models requires the predited marosopi stress-strain response and void growth behavior of the representative material volume to math the results obtained from detailed finite element models with expliit void representation. A series of arefully designed experiments ombined with finite element analyses of these speimens an also be used to alibrate the model parameters. As an example, a numerial proedure is proposed to predit dutile rak growth in thin panels of a 4-T3 aluminum alloy. The alibrated omputational model is applied to simulate rak extension in speimens having various initial rak onfigurations and the numerial preditions agree very well with experimental measurements. Keywords Dutile frature, Unit ell analysis, Porous plastiity model, Stress triaxiality, Lode angle 1. Introdution It is well-known that dutile frature in metalli alloys is a proess of void nuleation, growth and oalesene and this proess is strongly affeted by the stress state imposed on the material. Based on the frature mehanism, a straight-forward approah to simulate dutile failure proess is to model individual voids expliitly using refined finite elements [1-3]. A distint advantage of this approah is the exat implementation of void growth behavior. It provides an effetive method to study the mehanisms of dutile frature and to analyze the trend of frature toughness. However, due to sizeable differene between the harateristi length sales involved in the material failure proess and the dimensions of the atual strutural omponent, it is impratial to model every void in detail in struture failure analysis, espeially for situations involving extensive rak propagation. For this reason, various forms of porous material models have been developed to desribe void growth and the assoiated marosopi softening during the frature proess. The Gurson-Tvergaard-Needleman porous plastiity model [4-6], whih assumes voids are spherial in materials and remain spherial in the growth proess, has been widely used in modeling dutile failure proess and dutile rak extension. Gologanu, Leblond and Devaux [7, 8] extended the GTN model and derived a yield funtion for materials ontaining non-spherial voids. The GLD model an be applied to predit rak propagation in many proessed materials, suh as rolled plates. In literature, the stress triaxiality ratio, defined as the ratio of the mean stress to the equivalent stress, is often used as the sole parameter to haraterize the effet of the triaxial stress state on dutile frature. However, reent studies show that the Lode parameter must be introdued to distinguish the stress states having the same triaxiality ratio [3, 9-11]. In this study, we desribe a proedure to alibrate the material speifi porous plastiity model so that it an aurately apture the material behavior in the frature proess zone with the influene of the stress state. A numerial approah is proposed to predit dutile rak growth in thin panels of a 4-T3 aluminum alloy, where the GLD model is used to desribe the void growth proess and the material failure riterion is alibrated using experimental data. Model preditions are ompared with experimental data for frature speimens having various initial rak onfigurations. -1-

3 . Modeling the Material Behavior in the Frature Proess Zone Dutile alloys used in engineering strutures often ontain impurities suh as seond-phase partiles. Cavities often nuleate at relatively low stress levels due to frature or deohesion of the large inlusions. For the purpose of analysis, voids are assumed to be present in the material at the outset of loading. These voids enlarge with inreased plasti deformation and eventually oalese with the assistane of the nuleation and growth of seondary mirosopi voids. Therefore, material in the frature proess zone an be onsidered as an array of unit ells. Eah ell is a representative material volume (RMV) ontaining a void nuleated from the inlusion..1. Unit Cell Analysis A straight-forward approah to study the dutile frature mehanism as well as the effets of material properties and stress state on the material failure proess is to ondut the unit ell analysis of a representative material volume (RMV). As an example, Figure 1 shows a 1/8-symmetri finite element model for a ubi RMV ontaining a spherial void and Fig. 1 shows the three-dimensional stress state applied on the RMV. The material is assumed to obey a power-law hardening, true stress-strain relation with Young s modulus E=7.4 GPa, Poisson s ratio ν=.3, yield stress σ =345 MPa and strain hardening exponent N=.14. The initial void volume fration (volume of the spherial void / volume of the RMV) is taken as f =.. The initial size of the RMV is defined as X X X and the deformed lengths in the x-, y- and z-diretions are represented by X, Y and Z respetively. Σ y Σ 3 Σ 1 X / z x Figure 1. A one-eighth symmetri finite element mesh for the RMV ontaining a entered, spherial void. The stress state applied on the RMV. The stress state subjeted by the RMV is haraterized by two parameters, the stress triaxiality ratio (T) and the Lode angle (θ ) Σ + Σ + Σ Σ Σ Σ =, tanθ = 3Σ 3 Σ T (1) e ( Σ ) 1 where Σ e represents the von Mises equivalent stress. Here the numerial analyses are arried out --

4 using the finite element program ABAQUS [1], whih employs a finite strain, J plastiity theory within an updated Lagrangian formulation. The displaement boundary onditions on the outer surfaes of the RMV are presribed suh that the marosopi parameters T and θ are kept onstant during the entire deformation history. Faleskog et al. [13] and Kim et al. [9] provide the details of how to presribe the boundary onditions. o A ase of axisymmetri loading is onsidered first, where Σ Σ1 = Σ 3 ( θ = 3 ). Figure shows the variation of X with the marosopi effetive strain (E e ) of the RMV. As loading ontinues, X gradually dereases. But when the deformation reahes a ertain level, X stops dereasing and remains at a onstant value. This implies that further deformation takes plae in a uniaxial straining mode, whih orresponds to flow loalization in the ligament between adjaent voids. The shift to a marosopi uniaxial strain state indiates the onset of void oalesene. Detailed explanation of the uniaxial straining mode an be found in referenes Koplik and Needleman [14] and Kim et al. [9]. Here we use E to denote the marosopi effetive strain at the onset of void oalesene T= 1.5 T=/3 X / X.8 T=1 Σ e / σ 1. T= T=1.7.5 T=/ E e E e Figure. Variation of the deformed ell width in x-diretion with the marosopi true effetive strain of the ell revealing the shift to uniaxial straining. Marosopi true effetive stress versus true effetive strain of the void-ontaining RMV displaying the marosopi softening. The marosopi effetive stress versus effetive strain urve, Figure, provides an overview of the ompetition between matrix material strain hardening and porosity indued softening. As deformation progresses, a maximum effetive stress is reahed (indiated by the filled irle), and then Σ e dereases as strain-hardening of matrix material is insuffiient to ompensate for the redution in ligament area aused by void growth. As the marosopi effetive strain reahes E (indiated by the open irle), a rapid drop in marosopi effetive stress ours. As expeted, both the peak stress value and the value of E derease with the stress triaxiality ratio T, refleting the deease of dutility. -3-

5 In the later stage of the material failure proess, seondary voids often nuleate in the ligament between enlarged primary voids and rapid growth and oalesene of these seondary voids aelerates the final ligament separation. In our analyses, it is assumed that void nuleation is plasti strain ontrolled and follows a normal distribution proposed by Chu and Needleman [15]. The nuleated voids are regarded to be smeared in the material and the material behavior is governed by the GTN model. Figure 3 ompares the marosopi effetive stress versus effetive strain urves between models inluding and not inluding the seondary voids. Here several values of stress triaxiality ratio, T = 1/3, /3, 1, 1.5 and, are onsidered. The open irles denote the onset of oalesene for models where seondary voids are not taken into aount. The filled irles represent the onset of oalesene for models where nuleation, growth and oalesene of seondary voids are aounted for. It is lear that seondary voids signifiantly aelerate the void oalesene proess. To demonstrate the Lode angle effet on dutile failure, let the stress triaxiality ratio T be fixed and onsider a series of stress states orresponding to different θ-values. Figure 3 shows the variation of E with θ as T taking a fixed value of /3. Clearly the Lode angle has an important effet on E..5 f =. 1.6 T = /3. 1. Σ e / σ o T=1.5 T=. T=1. T=/3 T=1/3 E.8.4 no seondary voids with seondary voids E Θ (degree) e Figure 3. Comparison of the marosopi effetive stress versus effetive strain urves between models inluding and not inluding seondary voids. The parameters for nuleation of seondary voids are f N =.4, ε N =.1 and s N =.5 [15]. Variation of E with θ as T taking a fixed value of / Now onsider an array of T and θ values and perform unit ell analysis for eah ase. The variation of E with T and θ an be expressed by a funtion E (T,θ). Therefore, a dutile failure riterion for a given material an be established as Ee = E ( T, θ ) () where E e denote the marosopi effetive strain of the RMV. The RMV fails when E e reahes a ritial value dependent of its stress state haraterized by T andθ. -4-

6 Using expliit void representation, the void growth and oalesene mehanisms and the effets of the initial relative void spaing, void pattern, void shape and void volume fration on dutile frature toughness an also be studied diretly [16]... Porous Plastiity Models Various forms of porous plastiity models have been developed to desribe void growth in dutile solids and the assoiated marosopi softening, among whih the most famous model is due to Gurson [4] with the modifiation by Tvergaard and Needleman [5, 6]. The yield funtion of the GTN model has the form Σ 3 Φ = e osh Σ h + q 1 q f 1 f q σ 1 σ = where Σ e denotes the marosopi Mises effetive stress, Σ h represents the marosopi hydrostati stress, σ is the urrent flow stress of the matrix material, and f defines the urrent void volume fration. The evolution law for void volume fration is determined by requiring the matrix material to be plastially inompressible p ( - f ) & f & = 1 (4) E kk where E & is the trae of the marosopi plasti strain rate tensor. p kk The GTN model was derived for growth of spherial voids, but voids are often non-spherial in atual materials. The GLD model [7, 8], with the yield funtion given by Eq. (5), was derived to desribe the marosopi plasti response of dutile solids ontaining spheroidal voids C ' Σ h Φ = Σ + ησ + q( g + 1)( g + f ) osh ( g + 1) q ( g + f ) = hx κ (5) σ σ (3) where S is the shape parameter, denotes the von Mises norm, ' Σ is the deviatori stress tensor, Σ h is the generalized hydrostati stress defined by Σ h α ( Σ xx + Σ zz ) + ( 1 α ) Σ yy =, X is a tensor defined as X = ( / 3) e y e y ( 1/ 3) e x e x ( 1/ 3) e z e z, and (e x, e y, e z ) is an orthogonal basis with e y parallel to the axisymmetri axis of the void, and denotes tensor produt. The evolution equation for f is the same as Eq. (4) and derivations of the evolution equation for S an be found in Gologanu et al. [7, 8]. In order to simulate dutile frature proess, these porous plastiity models must be alibrated suh that the material behavior in the frature proess zone is aurately aptured. Calibration of these models requires the predited marosopi stress-strain response and void growth behavior of the representative material volume to math the results obtained from detailed finite element models with expliit void representation obtained from the unit ell analysis outlined in Setion.1. Faleskog et al. [13], Kim et al. [9] and Pardoen and Huthinson [17] desribe the proedures to -5-

7 determine the heuristi parameters in the GTN and GLD models as funtions of material flow properties, void parameters and the stress triaxiality. The porous plastiity models desribed above governs the behavior of the RMV during the void growth proess. As the marosopi effetive strain (E e ) reahes E, void oalesene ours and the RMV quikly loses its stress arrying apaity. We adopt the * f funtion, introdued by Tvergaard and Needleman [6], to aount for the effets of rapid void oalesene at failure. After E e reahes E, f is replaed by f * in the GLD model, where f * f, = f + K( f f ), f f f > f (6) In Eq. (6), f is the void volume fration at E e = E, K = ( f u f ) / f, and f u is the * f value at zero stress. Sine AQAQUS/Standard does not provide an element removal proedure, Eq. (6) is employed until f * * =. 99 f, after whih an exponential funtion is used suh that f gradually u approahes to f u (but an never reah f u ) to improve numerial stability. 3. Simulation of Crak Growth in Thin Panels of a 4-T3 Aluminum Alloy Dawike and Newman performed extensive frature tests on thin panels of a 4-T3 aluminum alloy, inluding tests of C(T), M(T), and multi-site damage (MSD) speimens [18, 19]. Figure 4 show the skethes of the frature speimens. The test data of our interest are from LT speimens with a sheet thikness of.3 mm. The speimens have very stiff guide plates (oated with Teflon tape) to onstrain out-of-plane (bukling) displaements. In the L orientation, the 4-T3 sheet material used in the experiments has a yield stress of 345 MPa, Young s Modulus of 71.3 GPa, and Poisson s ratio of.3. Quantitative metallographi analyses were performed to determine the inlusion volume fration, shape and average spaing. It is found that the inlusion volume fration (f ) is approximately., the average spaing between inlusions in the LT plane is about 5 μm, and in LT speimens, the inlusions an be approximated as prolate spheroids with the aspet ratio 4. To predit rak growth, the funtion ( T, θ ) needs to be determined. The results presented in E o o Setion.1 suggest that E is not sensitive to θ when θ is in the range - 3 θ. We perform finite element analyses of the frature speimens onsidered in this study and find the θ-values of o o the representative material volumes ahead of the rak front are in the range of -15 θ. Therefore, we neglet the θ-dependene for these speimens and assume E has the following funtion form -6-

8 E e βt = α (7) where α and β are two parameters need to be alibrated using experimental data. Two data points are needed to determine α and β. The tensile test provides one point. Figure 5 shows the 1/4 model for the tensile speimen and Figure 5 shows its experimental load-displaement urve. A sudden drop of the load-displaement urve suggests the onset of rak initiation. The stress and strain states for the ritial element (at the geometry enter of the speimen) at rak initiation are obtained through finite element analysis. The triaxiality T and strain ε f are alulated as.45 and.5 respetively. Substitution of these values into Eq. (7) yields a relationship between α and β,.45=αe.5β. The next step of the alibration proess seeks to math the model predited load versus rak propagation urve with the experimental measurements for the C(T) speimen. This step entails several finite element rak growth analyses of the C(T) speimen using different values of β. W a W L a L () W (d) W a b a b b L L a 1 a a 1 Figure 4. Frature speimens: C(T) speimen, M(T) speimen, () MSD speimen ontaining two raks, (d) MSD speimen ontaining three raks. The C(T) speimen has a width of 15 mm with a/w =.33, where a represents the initial rak length and W represents the speimen width. The quarter-symmetri finite element mesh has 7,4 eight-node, isoparametri solid elements (with redued integration). The mesh near the rak front has six layers with varying thikness to apture the stress gradient in the thikness diretion, where the thikest elements are at the symmetry plane. The elements diretly ahead of the rak front have uniform in-plane dimensions (L e = 5 μm) and are governed by the GLD model. All other elements follow J flow plastiity. Loading of the C(T) speimen is ontrolled by presribing a displaement -7-

9 on a rigid pin through the hole. Load (N) EXP FEA Figure 5. Finite element mesh for the tensile speimen. Comparison of the predited and experimental load-displaement urves. Figure 6 shows the omparison between the model predited load versus rak growth urve with the experimental measurements (two sets of experimental data) for different hoies of α and β, where the lines represent model preditions and the symbols denote experimental measurements. Here Δa represent the amount of rak growth measured at the free surfae. In the numerial model, the propagating rak front is defined by the elements whih have reahed the failure strain E. From Figure 6, it an be seen that the hoie of α =.93 and β = (solid line) results in a best fit to the experimental data. Therefore, these values are the alibrated values for α and β and will be used to predit rak growth in other frature speimens Displaement (mm) E =.8e -1.15T Ε =.161(T) Load (N) E =e T Ε =.84(T) E Ε =.1(T) =.93e -1.45T Δa (mm) Figure 6. Comparison of the model predited load versus rak growth urve with the experimental measured data (symbols) showing the hoie of α =.93 and β = (solid line) results in a best fit to the experimental data. -8-

10 The alibrated omputational model is now employed to predit the rak extension behavior of M(T) and MSD speimens. Three M(T) speimens with W = 3 mm and a/w ratios of.33,.4 and.5 are analyzed. The element size and arrangement in the region near the rak front are kept the same as used in the C(T) speimen. The nominal remote stress, σ R, haraterizes the loading for these speimens. Figure 7 ompares the omputed load versus rak extension responses with experimental measurements, showing very good agreement for all three ases. Figure 7 ompares the omputed load versus rak extension responses with experimental measurements for a MSD speimen ontaining three raks as shown in Figure 4(d). This speimen has the same width as the M(T) speimens. The enter-rak length is a = 1 mm. The two lead raks have the same length of a 1 = 1.5 mm. The tip-to-tip distane between the lead rak and the enter rak is b = 1.5 mm. The model predition aptures aurately the load versus rak extension urve. The usp on the predited load versus rak extension urve orresponds to the point when the lead rak and the enter rak link up σ R (MPa) σ R (MPa) Δa (mm) Exp.(a/W=.5) Exp.(a/W=.4) Exp.(a/W=.33) Num.(a/W=.5) Num.(a/W=.4) Num.(a/W=.33) Predition Experiment Δa (mm) Figure 7. Comparison of the model predited load versus rak extension responses (lines) with experimental measurements (symbols): M(T) speimens, MSD speimen ontaining three raks. 4. Conluding Remarks Based on the mehanism of dutile frature in metalli alloys, this paper desribes a method to predit rak growth in engineering strutures. To model extensive rak extension, homogenized porous plastiity models need to be adopted to desribe the material behavior in the frature proess zone and these models must be alibrated suh that the material behavior is aurately aptured. Unit ell analysis of the representative material volume reveals the strong effet of the stress state on the void growth and oalesene. Calibration of the porous plastiity models requires the predited marosopi stress-strain response and void growth behavior of the representative material volume to math the results obtained from detailed unit ell analysis. As an appliation, a numerial proedure is proposed to predit dutile rak growth in thin panels of a 4-T3 aluminum alloy. The material speifi GLD porous plastiity model is used to desribe the void growth proess and the failure riterion is alibrated using experimental data. The alibrated omputational model is then applied to predit rak extension in frature speimens having various initial rak onfigurations. The numerial preditions show good agreement with experimental -9-

11 measurements. Referenes [1] V. Tvergaard, J.W. Huthinson, Two mehanisms of dutile frature: void by void growth versus multiple void interation. Int J Solids Strut 39 () [] J. Kim, X. Gao, T.S. Srivatsan, Modeling of rak growth in dutile solids: a three-dimensional analysis. Int J Solids Strut 4(3) [3] J. Kim, G. Zhang, X. Gao, Modeling of dutile frature: appliation of the mehanism-based onepts. Int J Solids Strut 44 (7) [4] A.L. Gurson, Continuum of dutile rupture by void nuleation and growth: Part I-Yield riteria and flow rules for porous dutile media. J Eng Mater Teh 99(1977) -55. [5] V. Tvergaard, On Loalization in dutile materials ontaining spherial voids. Int J Frat 18(198) [6] V. Tvergaard, A. Needleman, Analysis of the up-one frature in a round tensile bar. Ata Metall 3(1984) [7] M. Gologanu, J.B. Leblond, J. Devaux, Approximate models for dutile metals ontaining nonspherial voids Case of axisymmetri prolate ellipsoidal avities. J Meh Phys Solids 41(1993) [8] M. Gologanu, J.B. Leblond, J. Devaux, J., Approximate models for dutile metals ontaining nonspherial voids Case of axisymmetri oblate ellipsoidal avities. J Eng Mater Teh 116(1994) [9] J. Kim, X. Gao, T.S. Srivatsan, Modeling of void growth in dutile solids: effets of stress triaxiality and initial porosity. Eng Fra Meh71(4) [1] L. Xue, T. Wierzbiki, Dutile frature initiation and propagation modeling using damage plastiity theory. Eng Frat Meh 75 (8) [11] K. Nahshon, J.W. Huthinson, Modifiation of the Gurson model for shear failure. Eur J Meh A/Solids 7(8) [1] SIMULIA, ABAQUS User s Manual (version 6.9), Providene, RI, 8. [13] J. Faleskog, X. Gao, C.F. Shih, Cell model for nonlinear frature analysis-i. Miromehanis alibration. Int J Frat 89(1998) [14] J. Koplik, A. Needleman, Void growth and oalesene in porous plasti solids. Int J Solids Strut 4(1988) [15] C. Chu, A. Needleman, Void nuleation effets in biaxially strethed sheets. J Eng Mater Tehnol 1(198) [16] X. Gao, T. Wang, J. Kim, On dutile frature initiation toughness: Effets of void volume fration, void shape and void distribution. Int J Solids Strut 4(5) [17] T. Pardoen, J.W. Huthinson, An extended model for void growth and oalesene. J Meh Phys Solids 48() [18] D.S. Dawike, J.C. Newman, Evaluation of frature parameters for predition residual strength of multi-site damage raking, in: Proeedings from the First Joint NASA/FAA/DoD Conferene on Aging Airraft, 1997, pp [19] D.S. Dawike, J.C. Newman, J.C., Residual strength preditions for multiple site damage raking using a three-dimensional finite element analysis and a CTOA riterion, in: T.L. Panontin, S.D. Sheppard (Eds), Frature Mehanis: 9 th Volume. ASTM STP 131, Philadelphia, PA, 1998, pp