A method for post-processing of ﬁnite element analyses to assess critical macroscopic voids is presented. One single, global analysis of the structure or component is carried out and then post-processed. By using a template model and the same constitutive model as for the global model, a damage scaling function is deﬁned once and then applied to the result from the global model. The result is a database with critical relative sizes for diﬀerent void rotations at the integrations points in the global model

Noury, P.

Unosson, M.

Veljkovic, M.

English

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XII International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS XIIE. On˜ate, D.R.J. Owen, D. Peric and B. Sua´rez (Eds)A METHOD FOR POST-PROCESSING OF FINITEELEMENT ANALYSES TO ASSESS CRITICALMACROSCOPIC VOIDSP. NOURY, M. UNOSSON AND M. VELJKOVICDepartment of Civil, Environmental and Natural Resources EngineeringLule˚a University of TechnologySE-971 87 Lule˚a, Swedene-mail: pourya.noury@ltu.se, web page: http://www.ltu.se/Key words: finite element analysis, pressure dependent damage, critical void, post-processingAbstract. A method for post-processing of finite element analyses to assess criticalmacroscopic voids is presented. One single, global analysis of the structure or component iscarried out and then post-processed. By using a template model and the same constitutivemodel as for the global model, a damage scaling function is defined once and then appliedto the result from the global model. The result is a database with critical relative sizesfor different void rotations at the integrations points in the global model.1 INTRODUCTIONThe presence of voids in metallic materials reduces its macroscopic mechanical strength.Voids in materials are a result of the manufacturing processes and the strength reductionis mainly due to two things: Increase in mean stress due to the reduction of solid ma-terial, i.e. less load carrying section area, and; local increase in strains and stresses onthe boundary of the voids. Voids can exist at different spatial scales. In an engineeringfinite element analysis the relative size of the voids are typically too small to be explic-itly modelled and discretized. Instead, the effects of the voids has to be incorporated inconstitutive equations and considered indirectly through test data on macroscopic prop-erties. Publication for ductile metals in this area seems to have started with for example[1], [2] and [3] and has since grown vast. Since such constitutive models treat voids collec-tively and in an averaging, homogenized fashion, they do not give any information aboutthe characteristics of individual voids. If one needs to formulate and supply specific ac-ceptance criteria, for example to be used for non-destructive testing of components, forindividual voids another route has to be followed. There are alternatives to be used inthis case, cf. for example [4], but a more general methodology is presented here which iseasier to automatize in a finite element context.1A method for post-processing of finite element analyses to assess critical macroscopic voids  1026P. Noury, M. Unosson and M. VeljkovicCommonly, in an engineering finite element analysis of a structure or component itis assumed that there are no extraordinary voids present in the material. Only smaller,i.e. for a given material and manufacturing process ordinary ones, whos effect on themechanical properties of the material are considered indirectly. Typically via tensile testdata in which these ordinary voids grow, coalesce and new ones nucleate until the specimenis completely fractured.In Fig. 1 a simple simulation of such a process, however without nucleation, is shown.From the pictures it can be seen how plastic strains are accumulated at the void boundariesand in bands between closest neighbours. When the stress dependent failure stress isreached, cracks are initiated and propagated until the plate is separated into two parts,i.e. it fractures.The focus of the presented study has been to develop methods with an engineering ap-proach, using finite element tools, to determine criteria for relative size and configurationof individual voids with elliptical shape in a steel structure. The expected void sizes arein the order of 10−3 mm.The basic idea here is to run one single standard finite element analysis of a structureof interest and corresponding loads, and then apply a scaling function to the result to findthe critical void configuration in a material/integration point that matches the weakestlink, in a macroscopic sense, of the void-free structure.2 COMPUTING THE SCALING FUNCTIONFirst, the scaling function needs to be determined. This function will be dependenton the constitutive model used, which needs to be the same used for the base model.However, once the scaling function has been determined for a given constitutive model itcan be applied to any problem.Here a standard isotropic elasto-plastic model with yield criteria according to Von Misesand piecewise linear deformation hardening is used, and a stress triaxiality dependentfailure model. The Young’s modulus and Poisson’s ratio were set to 210 GPa and 0.3,respectively. The constitutive parameters for the deformation hardening to necking werefitted against test data and the result is shown in Fig. 2.Due to lack of data for the specific material on the failure strain f at different levelsof stress triaxiality χ, typical values were chosen and shown in Tab. 1. However, thechoice of constitutive parameters here is not important since the purpose of the presentpaper is to present the method. It is only important that the constitutive model andparameter values can distinguish between compressive and tensile stress states, whichhere is embodied in the damage parameter.Damage D is assumed to be accumulated according to the linear evolution lawD˙ =˙pf(1)where the dot represents time derivative or increment. The material fails, i.e. a crackinitiates, when the damage D reaches unity.21027P. Noury, M. Unosson and M. VeljkovicTable 1: Failure strain at different levels of triaxialityχ −130 13231f 3 0.5 1 0.5 0.5The scaling function is determined by numerical analyses of a template model, seeFig. 3, for which the displacement based boundary conditions ∆x and ∆y, the ratio kbetween the minor and major axis of the elliptic void and the rotation v is varied. Herethe dimensions of the template model is chosen as h = 1 and r = 0.2. The intervals for thevariables are chosen as {k|0.1 ≤ k ≤ 1}, {v|0◦ ≤ k ≤ 180◦}, {∆x| − 0.002 ≤ ∆x ≤ 0.002}and {∆y|−0.002 ≤ ∆y ≤ 0.002}. Contact was defined for the void boundary to eliminatematerial penetration. Based on these intervals, a total of 3402 template models were setup and analyzed and in Fig. 5 the result is shown for the case v = 0◦ and k = 0.3. Thestrains x and y are the linear global strains over the template model, ie. for the casewhere no void is present.3 APPLICATION TO BOLTED CONNECTIONThe scaling function is applied to the results from a finite element analysis of a boltedconnection, see Fig. 6. In the analysis double symmetry is exploited when creating themesh. The weakest link in the base model is the bolt. So, the analysis is carried outuntil failure of the bolt, and it is this stress state in the plates to which we’ll apply thescaling function. Further, the strain state on the vertical symmetry plane is here assumedto be in a state of plane strain. When the bolt fails, some of the integrations points orelements in the plates has started to yield, and some are still in the elastic domain. Toexemplify the method of scaling the local solution, an element that is in elastic domainis chosen from the plate. The position of the chosen element and the damage scalingfunction corresponding to the elements stress state at bolt failure is shown in Fig. 7.The final step in the procedure is to choose a critical damage level. A conservativechoice would be D < 1. Since D = 0 corresponds to elastic domain and yielding in bothcompression and tension, it is recommended that a value larger than zero is choosen. Thisway we allow the pressure dependent damage evolution come into play, and we’re able tobetter separate compressive and tensile stress states. Here D = 0.025 is choosen as thetreshold value, and the result for the choosen element is plotted in Fig. 8. From this plotthe minimum ratio between the minor and major axis of the critical elliptic void can beseen, for different rotation of the major axis from the global x-axis.4 DISCUSSIONAn engineering method to post-process results from a finite element analysis in orderto determine critical size and rotation of elliptical voids has been developed. The methodallows for assessment of critical voids for any integration point, or element, in the model31028P. Noury, M. Unosson and M. Veljkovicand is far less costly than sub-modelling techniques. In the paper the method is appliedto one single element, but all elements in the model can be evaluated in the same mannerand the results visualized on the mesh as documentation to be used when performingnon-destructive testing such as ultrasonic testing or computed tomography scanning.Extension of the method to three dimensions is straight forward. In the paper it is therelative size of the voids that is determined. Further development of the method shouldinclude the determination of absolute void sizes, by for example relating the templatemodel to a characteristic element size. Or, if element sizes are small in comparisonwith the critical voids and boundary effects can’t be ruled out, some kind of movingrepresentative elementary volume. Finally, the effect of interaction between voids shouldalso be incorporated into the model.5 ACKNOWLEDGEMENTThe research leading to these results has received funding from the European Union’sSeventh Framework Programme managed by REA-Research Executive Agencyhttp://ec.europa.eu/research/rea ([FP7/2007-2013] [FP7/2007-2011]) under grant agree-ment no. 286603.REFERENCES[1] Tipper, C. The Fracture of Metals. Metallurgia (1949) 39:133–137.[2] McClintock, F. A. A criterion for ductile fracture by the growth of holes. Journal ofApplied Mechanics (1968) 35:363–371.[3] Rice, J. R. and Tracey, D. M. On the ductile enlargement of voids in triaxial stressfields. Journal of the Mechanics and Physics of Solids (1969) 17:201–217.[4] Guide to methods for assessing the acceptability of flaws in metallic structures. BS7910:2005, British Standards Institution, (2005).41029P. Noury, M. Unosson and M. VeljkovicFigure 1: Plastic straining and fracture of plate with voids at different stages of loading.51030P. Noury, M. Unosson and M. VeljkovicFigure 2: Deformation hardening.Figure 3: Template element for numerical computation of scale function.61031P. Noury, M. Unosson and M. VeljkovicFigure 4: Resulting damage D for v = 60◦, k = 0.15, x = y = 0.002.Figure 5: Scaling function for v = 0◦ and k = 0.15.71032P. Noury, M. Unosson and M. VeljkovicFigure 6: Equivalent Von Mises stress (MPa) at bolt failure.81033P. Noury, M. Unosson and M. VeljkovicFigure 7: Damage for an element in the finite element model when scaling function is applied.91034P. Noury, M. Unosson and M. VeljkovicFigure 8: Critical and non-critical voids for D = 0.025.101035

A method for post-processing of finite element analyses to assess critical macroscopic voids

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A method for post-processing of finite element analyses to assess critical macroscopic voids

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