On the Proportionality of Damage Rule in Finite Element Simulations of the Ductile Failure

Transcription

1 Advanced Materials Research Submitted: ISSN: , Vol. 98, Acceted: doi:.48/ Online: Trans Tech Publications, Switzerland On the Proortionality o Damage Rule in Finite Element Simulations o the Ductile Failure František Šebek,a, Jiří Hůlka,b, Petr Kubík,c and Jindřich Petruška,d Brno University o Technology, Brno, Technická 896/, Brno, Czech Reublic a y7598@stud.me.vutbr.cz, b hulka@c-box.cz, c m3etr@gmail.com, d etruska@me.vutbr.cz Keywords: ductile ailure, damage rule, uncouled model, exlicit. Abstract. Extended Mohr Coulomb criterion which is uncouled, thereore the lasticity is not inluenced by the damage, was selected and calibrated. Exeriments on the aluminum alloy 4-T35 were carried out. Further, the imact o the damage rule roortionality was investigated. The damage exonent was sequentially held, and 3, resectively. Thereore, the inluence o the damage rate on the damage ailure in inite element simulations was examined. Numerical simulations o the tensile test o smooth and notched cylindrical secimens and lat grooved secimen were erormed. It is also shown that there is inability to redict slant racture by uncouled models or lat grooved secimen and in the inal hase o tensile tests o cylindrical secimens. Introduction The ductile ailure rediction has been still a great challenge not only or mechanical engineers. It also could be imortant or etroleum or civil engineering. So it really has a considerable imact in many braches o engineering and in many ractical industrial alications. Number o robust ailure models and damage aroaches has been develoed so ar. Paer is ocused on the emirical criteria which can also be interreted as henomenological. The ioneer work in this ield was ublished by Cockrot and Latham []. When redicting ductile ailure, one has to know the lasticity o examined material. In case o uncouled ailure models, the lasticity is not inluenced by the damage rocess and the lastic low remains during straining the same as the low curve identiied rom standard tensile tests, until the racture occurs. The other art, when modeling the ductile ailure, is to have a reliable ductile ailure model. However, this aer does not deal with assessing the reliability or redictive accuracy o such models. This roblem has been discussed in number o ublications and it is not the aim o this aer. One o the crucial arts o ductile ailure model could be its damage rule, so the inluence o the damage rate, controlled through damage exonent, is studied hereinater. Aroach to Ductile Failure Modeling Wierzbicki et al. [] carried out iteen racture tests including various secimens. They obtained low curve using tensile tests o smooth secimens. The stress-strain relationshi was described by n σ = A, () where σ, A, and n are the equivalent Mises stress, stress amlitude, equivalent total strain and strain hardening exonent, resectively. Bai and Wierzbicki [3] introduced a new model o metal lasticity with ressure and Lode deendence. By suitable setting o material constants, the model is reduced either to von Mises or Tresca yield criterion. Later, Bai and Wierzbicki [4] modiied Mohr Coulomb ailure criterion using the revious lasticity model and introduced Extended Mohr Coulomb criterion as another uncouled henomenological ailure model. This is an oen access article under the CC-BY 4. license (htts://creativecommons.org/licenses/by/4./)

2 9 Modern Technologies or Engineering, Alied Mechanics and Material Science Assuming material constants o lasticity model [3] in order to obtain and adot von Mises yield criterion, the racture strain o ailure model [4] concerning two material constants, c and c, is exressed as n A c + π π = cos θ + c sin c 3 6 η + θ 3 6, () where θ and η are the normalized third invariant o the deviatoric stress tensor and stress triaxiality, resectively. Calibration constants o the ailure model discussed above are obtained through non-linear least square method which can be generally written in vector terms as min x where ( x) ( ) ( x) ( x) + ( x) + + ( x) = min k, (3) x and x are a unction that returns a vector value and vector, resectively. In our case, k reresents the number o racture tests. The last art o ductile ailure model is the damage rule. Xue [5] derived the damage evolution law as a ower unction m d dd = m, (4) where m and are the damage exonent and equivalent lastic strain, resectively. Calibration o Material Constants At irst, material constants related to lasticity were taken over, A = 744 MPa and n =. 53, according to Eq. () rom []. These were not used only in ductile ailure model in Eq. () but also as inut into inite element simulations. Further, the racture strains and average values o the stress triaxiality and normalized third invariant o the deviatoric stress tensor were taken over []. Average values, used in calibration rocess, were determined using numerical integration η ξ av av = = ηd, (5) ξd. (6) Following exression was used or conversion rom the normalized third invariant o the deviatoric stress tensor to normalized Lode angle θ = arccos( ξ ), (7) π where ξ is the normalized third invariant o the deviatoric stress tensor. The material constants o Extended Mohr Coulomb criterion, c =. 6 and c = in Eq. (), were obtained through non-linear least square method using Eq. (3). The racture enveloe according to these constants and taken over values o racture strain is deicted in Fig..

3 Advanced Materials Research Vol Bai and Wierzbicki [4] obtained similar results. Slight variance is due to slightly dierent values o the stress amlitude and strain hardening exonent which were used in [4]. Finite Element Simulations Fig. Fracture enveloe o Extended Mohr Coulomb criterion The inite element simulations were realized within Abaqus/Exlicit commercial code via the user subroutine VUMAT. Numerical simulations o tensile tests o smooth and notched cylindrical secimens and lat grooved secimen were conducted (Fig. ). The axisymmetric condition is generated in cylindrical secimens and lane strain in the lat grooved secimen. Smooth secimens had 9 mm diameter. Two tyes o notched secimens had notch radii 4 and mm and both 8 mm diameter in minimal cross-section. Flat grooved secimens had thickness 3 mm and width.5 mm. The gauge length was 5.4 mm or all secimens [, 6]. Fig. From let, geometry o lat grooved secimen, smooth secimen and notched secimens Fields o damage or each secimen, immediately beore crack initiation and ater searation, and sequentially or damage exonent, and 3, in order to gain linear, quadratic and cubic damage evolution, resectively, are deicted in Figs. 3-6.

4 9 Modern Technologies or Engineering, Alied Mechanics and Material Science m = m = m = 3

5 Advanced Materials Research Vol In Fig. 7 are shown histories o damage evolution and damage rate. Damage arameter [-] Summary m = m = m = Equivalent lastic strain [-] Damage rate [-] Flat grooved R = 4 mm R = mm Smooth secimen Equivalent lastic strain [-] Fig. 7 Histories o damage evolution (let) and damage rate (right) The roortionality o damage rule was examined through changing the damage exonent. When the damage exonent equals to unity, the damage arameter is roortional to the equivalent lastic strain and thereore the damage rate is constant. It can be clearly seen that localization roblem at lat grooved secimen, when the lane strain condition is generalized, is not distinct. The localization is signiicant at cylindrical secimens, esecially at the smooth one. Slant racture at inal stage is redicted only or notched secimen with notch radius mm. One can see that slant cracks roagated with rising damage exonent, but stoed and the major lat crack grew urther until searation. Finally, there was some localization eect but not strong enough to govern the slant racture where it was observed exerimentally. Acknowledgment This work is an outut o cooeration between grant roject FSI-S-4-3 and NETME Centre, regional R&D centre built with the inancial suort rom the Oerational Programme Research and Develoment or Innovations within the roject NETME Centre (New Technologies or Mechanical Engineering), Reg. No. CZ..5/../. and, in the ollow-u sustainability stage, suorted through NETME CENTRE PLUS (LO) by inancial means rom the Ministry o Education, Youth and Sorts under the National Sustainability Programme I. Reerences [] M. G. Cockrot, D. J. Latham, Ductility and the Workability o Metals, J. Inst. Metall. 69 (968) [] T. Wierzbicki, Y. Bao, Y.-W. Lee, Y. Bai, Calibration and evaluation o seven racture models, Int. J. Mech. Sci. 47 (5) [3] Y. Bai, T. Wierzbicki, A new model o metal lasticity and racture with ressure and Lode deendence, Int. J. Plast. 4 (8) [4] Y. Bai, T. Wierzbicki, Alication o extended Mohr Coulomb criterion to ductile racture, Int. J. Fract. 6 () -. [5] L. Xue, Damage accumulation and racture initiation in uncracked ductile solids subject to triaxial loading, Int. J. Solids Struct. 44 (7) [6] Y. Bao, Prediction o ductile crack ormation in uncracked bodies. Ph.D. thesis, Massachusetts Institute o Technology, 3.