EFFECT OF STRAIN RATIO ON STRESS TRIAXIALITY SUBJECTED TO MODE I FRACTURE

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1 EFFECT OF STRAIN RATIO ON STRESS TRIAXIALITY SUBJECTED TO MODE I FRACTURE 1 RUCHIN KACKER, 2 SHAILENDRA SINGH BHADAURIA, 3 VISHAL PARASHAR 1,2 Dr. B.R. Ambedkar National Institute of Technology Jalandhar, Punjab India 3 Maulana Azad National Institute of Technology Bhopal, M.P India ruchinkacker@gmail.com, bhadauriass.nit.ac.in, parashar17@rediffmail.com Abstract- Most fabricated metal shapes in several products and components used in industry have anisotropic properties. This study deal with the triaxiality based mathematical models developed for Mode I fracture using Hill s von Mises' yield criterion, for an anisotropic material having orthotropic symmetry. R (ratio of width strain to thickness strain) in case of anisotropy is the utility factor therefore on the basis of Rvalue the model for stress triaxiality is formulated and then compared to isotropic model in plane stress and plain strain condition. Criticality of M (stress triaxiality) is validated to plastic zones at each step. Keywords- R (strain ratio), Triaxiality, von Mises Yield criterion, Hill-von Mises yield criteria, Mode-I fracture I. INTRODUCTION Theories which govern crack propagation are priory based on fundamental concepts of stress, strain, strain energy density, displacement at the crack tip. As far as anisotropy in materials is concerned the prediction of crack propagating at a particular angle as depicted by a particular theory differs. This problem of crack propagation is of particular significance to linear elastic fracture mechanics to know crack initiation and further its growth under different set of loading conditions. Anisotropic materials find their variety of uses in blades of turbine, sheet metals, MEMS, NEMS etc. Along with various parameters of fracture mechanics such as stress intensity factor, crack tip opening displacement, fracture toughness, Energy release rate etc. stress triaxiality is also an important parameter of ductile fracture. Several studies have been concerned with triaxiality, strain ratio and anisotropic materials, illustrated brefly as follows. Assuming a plastically anisotropic matrix material finite strain analyses has been conducted under strain conditions and it was revealed that failure occurred as a result of sudden stress drop. It has been also observed that geometric anisotropy may develop the overall ductility however with the snag of considerable drop in load carrying capacity after primary debonding. A phenomenological yield function has been proposed to represent the plastic anisotropy of aluminium sheets. Anisotropy has been represented by twelve parameters in form of two fourth order symmetric tensor. The function is found to represent the anisotropy of 2024 aluminium sheets alloy along with other anisotropic materials. Investigations on 2024-T351 aluminium alloy have been performed using micromechanics based damage model that accounted for the effect of void aspect ratio and void distribution. The effects of void shape and void spacing on fracture behaviour were explained by means of finite element cell calculations. A macroscopic ductile fracture criterion has been proposed based on model of micro-mechanism analysis of nucleation, growth and shear coalescence of voids from experimental observation of fracture surfaces, model endows a changeable cut-off value for the stress triaxiality to represent effect of microstructures, the Lode parameter, temperature, and strain rate on ductility of metals.model is also used to construct fracture loci of AA 2024-T351. The retical background of fracture phenomena and a failure strain criterion in marine structural steels has been proposed for EH36, one of the most popular polar class steels, based on experimental and numerical investigations Instead of using local stress triaxiality, critical strain energy concept and corresponding average failure stress triaxiality has been introduced. It is proved that EH36 high strength steel well obeys a failure strain curve with 100% critical energy in a limited average failure stress triaxiality zone from 0.5 to 1.0 [5].Effect of stress state on the damage behaviour of ductile metals has been investigated. The continuum damage model has been generalized to take into account the effect of stress state on damage criteria as well as on evolution equations of damage strains. Different branches are considered corresponding to various damage mechanisms depending on stress intensity, stress triaxiality and the Lode parameter. To be able to get more insight in the complex damage and failure behaviour additional series of three-dimensional micro-mechanical numerical analyses of void containing unit cells have been performed. The numerical results are used to show general trends, to develop equations for the damage criteria, to propose evolution equations of damage strains, and to identify parameters of the continuum model. Anisotropic nature of mixed mode (I+II) crack tip plastic core region and crack initiation has been investigated using an angled crack plate problem under various loading conditions. In addition, R-criterion has also been extended the for crack initiation proposed by the other authors for isotropic materials and showed 53

2 effect of Hill s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle for both plane stress and plane strain conditions at the crack tip.the study showed a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory [7].Hill s 1979 anisotropic yield criterion were examined. Examination revealed that for cases I, II and III, there are combinations of m and R for which the yield loci are outwardly concave or even unbounded. For case I all loci are concave unless m=2.for case II and III, the combinations of m and R which lead to concavity and unboundedness were calculated. Case IV and Hosford s criterion have no problem as long as m 1. This study is related to anisotropic sheets under plane stress condition under mode I fracture. Comparison of model for stress triaxiality for anisotropic plane stress condition has been done, first with stress triaxiality for isotropic plane stress condition and sequentially then to isotropic plane strain condition. II. METHODOLOGY Mathematical formulation has been carried out for M (stress triaxiality) for the anisotropic material having Orthotropic symmetry and subjected to mode-i fracture, assuming that the yield stress σ y to be equal in the plane of sheet and also the thickness stress 3 σ is assumed to be negligible of a highly textured sheet. Then on the basis of R (strain ratio) value which is an important parameter for anisotropy this stress triaxiality is compared with the stress triaxiality for isotropic ductile material in plane stress and plane condition for mode-i condition. Critical triaxiality has been found out in each of the cases. Polar plots in support have been shown for the same. III. MATHEMATICAL MODELLING 54

3 IV. FIGURES AND DRAWINGS Fig.1A Variation of M(stress triaxiality)with angle of inclination from crack tip, R(strain ratio) varying from Fig.1B Variation of stress triaxiality with R (strain ratio) at various inclinations from crack tip

4 Fig 2 Polar plot of anisotropic plane stress condition for R varying from Fig. 5 Polar plot for plane strain showing effect of anisotropy from of R at Poisson s ratio 0.2 Fig. 3A Variation of stress triaxiality with inclination from crack tip for Poisson s ratio from Fig.6 comparison of Polar plots of 0.1* R for plane stress and all Poisson s ratio V. RESULTS AND DISCUSSION Fig. 3B Variation of triaxiality with Poisson s ratio for different increasing inclinations from crack tip. Fig. 4 Comparison of M (stress triaxiality) at 0.3* R for υ varying Fig 1 show that at zero degree the value of triaxiality reaches its criticality at various values of R (strain ratio) and as the magnitude of R-value increase the critical value go on increasing. This fact has been validated from Fig 2 i.e. as the R-value increase the radius of plastic zone, at zero degree angles of nclination decreases. Secondly the effect of Poisson s ratio has also been shown on stress triaxiality in Fig 3 i.e. as Poisson s ratio increase criticality increase and this fact has been validated again through minimizing of plastic zone shown in Figs. 5 and 6. Thirdly, R and R are no more a mere constant like R because in plain strain the effect of 3 has been incorporated in R. R is now represented as R as eq. (20) suggest including Poisson s ratio. All equation in plain strain condition are so formulated that as υ attain the zero value they are similar to plane stress condition i.e. the behaviour 56

5 of plastic zone of isotropic plane stress is same that of isotropic plane strain or viceversa. CONCLUSION Though anisotropy in plain strain condition is not common henceforth the empirical relations regarding yielding are not present however in present study, mathematical equations of anisotropic plane strain condition id obtained using its fundamental relation in plane stress condition and isotropic plain strain condition while considering an important ductile fracture parameter stress triaxiality. REFERENCES [1] B. N.Legarth, Effects of geometrical anisotropy on failurein a plastically anisotropic metal, Engineering Fracture Mechanics, vol. 72, pp , June [2] F. Bronand J. Bessona, A yield function for anisotropic materials application to aluminium alloys, International Journal of Plasticity, vol. 20,pp , June [3] D. Steglich, W. Brocks,,J.Heerensand T.Pardoen, Anisotropic ductile fracture of Al 2024 alloys, Engineering Fracture Mechanics, vol. 75, pp , April [4] Lou Yanshan, J.W. Hoon and H. Huh, Modelling of shear ductile fracture considering a changeable cut-off value for stress triaxiality, International Journal of Plasticity, vol. 54, pp , August [5] J Choung, C.S. Shim and H.C. Song, Estimation of failure strain of EH36 high strength marine structural steel using average stress triaxiality, Marine Structures, vol. 29, Issue 1, pp. 1-21,December 2012 [6] M. Brunig, S. Gerke and V.Hegerbrock, Micro-mechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage,international Journal of Plasticity, vol.50,pp.49-65,november [7] M.A. S. Khanand K. M. Khraisheh, The anisotropic Rcriterion for crack initiation, Engineering Fracture Mechanics, vol. 75,pp , April 2008 [8] Y. Zhu, B. Dodd, R. M. Caddell and W. F. Hosford, Convexity restrictions on non-quadratic on anisotropic yield criteria, Int. J. Mech. Sci. vol. 29 no. 10/11, pp ,