CHARACTERIZATION OF MATERIAL DUCTILITY BY MICROVOID NUCLEATION PARAMETERS

Transcription

1 Proceedings o the 9 th Risø International Zhang, Ødegård, Symosium Thaulow on Materials Science: Modelling o Structure and Mechanics o Materials rom Microscale to Product. Editors: J. V. Carstensen, T. Leers, T. Lorentzen, O. B. Pedersen, B. F. Sørensen and G. Winther. Risø National Laboratory, Roskilde, Denmark 998. CHARACTERIZATION OF MATERIAL DUCTILITY BY MICROVOID NUCLEATION PARAMETERS Z. L. Zhang, J. Ødegård and C. Thaulow SINTEF Materials Technology, N-748 Trondheim, Norway ABSTRACT The ductility and racture toughness o a material vary with the change o geometry and can not be directly transerred rom one geometry to another. For olycrystalline metals, ductile racture occurs most oten as a consequence o nucleation, growth and coalescence o microvoids. It thereore becomes natural to link material racture behavior to the micro-mechanical arameters, which describe the evolution o microvoids, through a micro-mechanical model. Recently, a comlete Gurson model has been roosed by the authors. In the comlete Gurson model, both void nucleation and growth characterized by the Gurson model, and void coalescence characterized by Thomason s lastic limit load model are included. With the comlete Gurson model, void coalescence is a natural outcome o lasticity develoment, and ductile racture is exlicitly linked to the void nucleation arameters. A method has been roosed to characterize material s ductility according to nucleation arameters.. DEFORMATION AND DUCTILE FRACTURE Material deormation behavior can be characterized by its true stress and strain curve, which includes both the yield strength and hardening exonent. The true stress strain curve is geometry indeendent and can be transerred rom a standard tensile test to a comlicated geometry. The conventional method to characterize ductile racture is to use macro racture arameters such as ductility (or un-recracked secimen) or racture toughness (or cracked secimens). However, unlike the deormation arameters, macro racture arameters are geometry deendent and can not be directly transerred rom one geometry to another. It has been generally understood that the conventional racture mechanics works only in a limited number o cases. For olycrystalline metals, ductile racture occurs most oten as a consequence o nucleation, growth and coalescence o microvoids. It is thereore natural to link 565

2 Characterization o ductility by void nucleation arameters material racture behavior to the arameters that describe the evolution o microvoids through a micro-mechanical model. Several micro-mechanical models have been develoed in the ast or modeling ductile racture and one o the best known models is due to Gurson (Gurson 975). The Gurson model can well redict void growth and material sotening resulting orm the resence o voids, but it has no intrinsic ability to redict void coalescence. Recently, a comlete Gurson model in which the Gurson model is used or describing void nucleation and growth, and Thomason s lastic limit load model is used or characterizing void coalescence has been resented (Zhang 998). It has been shown that rediction o void coalescence by the comlete Gurson model is very accurate comared with FE analysis o cell models (Zhang 998). The conventional measure o ductility is racture strain. With the comlete Gurson model, the deendence o ductility on geometry, i.e. notch radius, can be uniquely described by void nucleation arameters. The nucleation arameters can be determined rom a series o and notched tensile tests. The nucleation arameters can then be used to characterize material s ductility and to redict ductile racture.. A COMPLETE GURSON MODEL In the literature, several void growth models have been roosed. The models describe void growth rate as a unction o remote alied lastic strain rate and current stress triaxility, neglecting the eedback o microvoids on material s load carrying caacity (Rice and Tracey 969). In 975, Gurson resented a model, which can redict the void growth and the sotening eect due to voids. The Gurson model has been urther modiied to take into consideration o the eect o void interaction (Tvergaard 98,99) and void shae (Søvik 996). However, a common eature o the Gurson model and other void growth models is that no intrinsic void coalescence criterion is included. The models can calculate void growth but another ailure criterion should be used, such as the critical void volume raction / void growth ratio or the critical value o the distance between crack ti and the neighboring void. Recent numerical and exerimental results have shown that the critical void volume raction is deendent on the initial value o the void volume raction and also on the stress triaxility when the initial void volume raction is large (Zhang 998). Thomason has develoed a ductile racture model (Thomason 99), and argues that lastic localization and void coalescence aears when the lastic limit load o the inter-void matrix material has reached. The lastic limit load, reresented by, σ is very large when the void radius is small comared with the inter-void dimension, and is decreasing when the lasticity develos and the void radius increases. Finally, when the alied microscoically homogenous stress, σ, is equal to lastic limit load stress, σ Homogenous, racture by void coalescence will occur. 566

3 Zhang, Ødegård, Thaulow By integrating the Gurson model and the Thomason model together and assuming voids are always sherical or cylindrical, a comlete Gurson model which models ductile racture as the comlete rocess o void nucleation, growth and coalescence is obtained. Detailed descrition and veriication o the comlete Gurson model can be ound (Zhang 998, Zhang and Hauge 998, and Zhang and Niemi 995). The comlete Gurson model is schematically shown in Fig.. Homogenous σ σ Homogeneous deormation by the Gurson model deormation z x σ σ z r x R R Fig. The comlete Gurson model: void nucleation and growth by the Gurson model a) and void coalescence by the Thomason model b). Void is always assumed to be sherical (3D) or cylindrical (D lane strain) (Zhang 998). With the comlete Gurson model, void coalescence is not directly a material roerty but becomes a natural outcome o lasticity develoment ollowing the void nucleation. Thereore, in the rocess o ductile racture, void nucleation is the most critical event, and void growth and coalescence are the resonse o the nucleated voids to the alied lasticity and stress triaxility. According to the comlete Gurson model, ductile racture is uniquely controlled by materials void nucleation arameters. Void nucleation belongs to material intrinsic roerties. 3. VOID NUCLEATION MODELS Engineering alloys usually contain inclusions, or examle, manganese sulides and aluminium oxides in steels, and second-hase articles, or examle, carbides in steels and intermetallic hases in aluminium alloys. Void nucleation in general deends on article strength, size, shae and the hardening exonent o the matrix material. The nucleation mechanism can be strain controlled, or stress controlled where the hydrostatic tension stress lays a role (Zhang 998). For strain controlled criterion, the contribution o void nucleation to the increase o void volume raction, d, can be written: d = ( ) d () where is the void nucleation intensity and is the equivalent lastic strain. For many engineering alloys, the ollowing two simle nucleation models can be used. The 567

4 Characterization o ductility by void nucleation arameters irst is a cluster nucleation model characterized by the initial void volume raction, where all the voids are assumed to nucleate at the beginning o lastic deormation. This model can be alied to materials where large inclusions exist or micro cracks are resent in the material. Another nucleation model is called continuous nucleation model, where it is assumed that the amount o nucleated voids er increment o lastic equivalent strain is a constant, which is denoted as A. These two models are schematically shown in Fig.. a) b) A Fig. Void nucleation models, the cluster nucleation model characterized by the initial void volume raction a), and the continuous nucleation model characterized by a constant A b). The arameters which describe the void nucleation models are in general called micromechanical arameters. It should be noted that real nucleation mechanisms are very comlicated, and the roosed models are aroximate ones. The tyical eature o the micro-mechanical arameters is that they can not be directly measured and the directly measured arameters can not be used. Numerical methods must be used or determination o void nucleation arameters. The relation between the ductility o a material, to its nucleation arameters,, can be written: = (, G), () where G reresents the geometry o a tensile secimen. 4. DUCTILITY DIAGRAM With the comlete Gurson model, the ductile racture is exlicitly related to the void nucleation arameters, Eq. (). For given tensile test results, dierent void nucleation models and arameters can be tested. The one which its best with the test results is chosen as the real one. A series o tensile tests with and notched secimens are suggested or itting the void nucleation arameters. When measured racture strains are lotted with the notch radius or stress triaxility at the centre o secimens, a so-called ductility diagram is obtained. Fig. 3 shows the ductility diagram or a ieline steel X65 (Taule 997). The diameter o the tensile secimens is 6 mm. Smooth secimen and secimens with notch radius, R=.,.,.8 and.4 mm were used. In the material, the largest article diameter is 5.6 µ m and the number ercentage o these articles is less than %. Most o the articles have a diameter about. µ m. The 568

5 Zhang, Ødegård, Thaulow two nucleation models shown in Fig. have been tested. Fig. 3 shows that the cluster nucleation model does not it the ductility diagram. The best itting can be obtained by considering two nucleation mechanisms both or large inclusions and small articles (Taule 997). As an aroximation, the continuous model with A=.8 gives a good it to the test results. The continuous nucleation model can be considered to reresent a mixture o both the large and small articles or this material. An Al-4.3%Si aluminium alloy has also been tested. Secimens with 4mm diameter were used. Same notch radius as the steel has been used or the notched secimens. The itting results o the two nucleation models are shown in Fig. 4. It has been ound that the continuous nucleation model, Fig. 4a, does not it with the material, and the cluster nucleation model, Fig. 4b, with =.% its very well, Fig. 4b. Metallurgical examination shows that micro-deects acting as microvoids exist in the material rior to testing due to extrusion o the material. The measured initial volume raction o microvoids is about.% which is close to the initial void volume raction itted rom Fig. 4b (Ødegård, 997). On other hand, it can be concluded that the ductility o this material is rimarily controlled by the micro-deects. 3,5,5 =.8 =.5 =. R=.4 mm 3,5,5 A=.5 A=.8 A=.5 R=.4 mm,5,5,,5,,5, Stress triaxility in the secimen centre T,,5,,5, Stress triaxility in the secimen centre T a) b) Fig. 3 Ductility diagrams or a X65 ieline steel, a) cluster nucleation model and b) continuous nucleation model b).,5,,9,6,3 A=.5 A=. R=.4 mm,5,,9,6,3 =. =. R=.4 mm,,,5,,5, Stress triaxility in the secimen centre T a) b) Fig. 4 Ductility diagram or an Al-4.3%Si alloy, a) continuous nucleation model; and cluster nucleation model b).,,,5,,5, Stress triaxility in the secimen centre T 569

6 Characterization o ductility by void nucleation arameters 5. DISCUSSIONS AND CONCLUSIONS In this aer, material ductility has been exlicitly linked to the void nucleation arameters, A or. The deendence o ductility on geometry is comletely described by the nucleation arameters. The established void nucleation arameter can be used or qualiication o material and material selection, and or the rediction o ductile racture o un-recracked secimens and the racture toughness o cracked secimens. For ractical urose, the selection o nucleation model can be determined rom the racture strain ratio o secimen to a reresentative notched secimen. The cluster nucleation model results in high ratio and the continuous nucleation is the oosite. One o the imortant asects in the alication o the Gurson model based aroach is the determination o micro-mechanical arameters. The comlete Gurson model and the roosed ductility diagram rovide a method or determining the micromechanical arameters. With the comlete Gurson model, the real void nucleation model can be obtained by using the ductility diagram. It should be mentioned that dierent void nucleation models can give same rediction at one stress triaxility while the rediction is very dierent at another stress triaxility, see Fig. 5. It is necessary to test the itting o void nucleation models and arameters at dierent stress triaxility, and thereore a ductility diagram should be used. Dierent materials ollow dierent nucleation models and will have dierent nucleation arameters. The nucleation arameters deend on materials microstructure. The ongoing research activities ocus on the link between the nucleation arameters and materials microstructure, and eventually to the manuacturing arameters. 3,5,5,5 =. A=.8,,5,,5, Stress triaxility in the secimen cent T,5,5 =.5 A=.85,,5,,5, Stress triaxility in the secimen cent T Fig. 5 Dierent nucleation models redict nearly the same ductility at one stress triaxility case but redict very dierent ductility at other stress triaxility cases. REFERENCES 57

7 Zhang, Ødegård, Thaulow Gurson A. L. (975). Plastic Flow and Fracture Behaviour o Ductile Materials Incororating Void Nucleation, Growth And Coalescence. Ph.D Dissertation, Brown University. Thomason P. F. (99). Ductile Fracture o Metals. Pergamon Press, Oxord. Rice J. R. & Tracey D. M. (969). On the ductile enlargement o voids in triaxial stress ields. J. Mech. Phys. Solids 7, -7. Søvik O. P. (996). Numerical Modeling o Ductile Fracture - A Damage Mechanics Aroach, Ph.D thesis, Norwegian University o Science and Technology. Taule Å. (997). Quantiication o ductile crack growth in ielines by means o damager mechanics calculations, MSc thesis, NTNU. Tvergaard V. (98). On Localization in Ductile Materials Containing Sherical Voids. Int. J. Fract. 8, Tvergaard V. (99). Advances in Alied Mechanics. J. W. Hutchinson & T. Y. Wu (eds.), Academic Press, Zhang, Z. L. (998). A comlete Gurson model. In: Nonlinear Fracture and Damage Mechanics edited by M. H. Aliabadi, Comutational Mechanics Publications. Zhang Z. L. & Niemi E. (995). A New Failure Criterion or the Gurson-Tvergaard Dilatational Constitutive Model. Int. J. Fract. 7, 3-334, 995. Zhang Z. L., Hauge M. (998). On the Gurson arameters, Fatigue and Fracture Mechanics: 9 th Volume, ASTM STP 3, T. L. Panontin and S. D. Sheard, Eds, American Society or Testing and Materials. Ødegård J. (997). Quantiication o damage arameters and void voluem raction. SINTEF Reort, STF4 A