A proper set of material parameters is one of the most important aspects for a successful simulation of metal forming processes. Several issues must be observed when choosing the constitutive relation and corresponding material parameters, amongst which the most important are: (i) the magnitude of the plastic deformation of the target forming operation must be contemplated by the parameters of the constitutive model, (ii) possibility of failure prediction in fracture-free materials, and (iii) accurate prediction of geometrical changes caused by plastic deformation. Within this framework, the present article discusses techniques to obtain constitutive parameters of a Lemaitre-type material model. The strategy requires compliance of multiple tensile tests with specimens prepared according to diﬀerent technical standards. Parameter identiﬁcation is regarded as an inverse problem and solved using optimization methods

Tomiyama, M.

Vaz Jr, M.

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IS-Computational Strategies for Metal Cutting & Forming OperationsFurther Studies on Identific tion of Inelastic Parameters for Damaged MaterialsM. Vaz jr. and M. TomiyamaXV International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS 2019E. On˜ate, D.R.J. Owen, D. Peric, M. Chiumenti & Eduardo de Souza Neto (Eds)FURTHER STUDIES ON IDENTIFICATION OF INELASTICPARAMETERS FOR DAMAGED MATERIALSM. VAZ JR. AND M. TOMIYAMADepartment of Mechanical EngineeringState University of Santa Catarina (UDESC)Campus Universita´rio Prof. Avelino Marcante, 89219-710 Joinville, Brazile-mail: miguel.vaz@udesc.brKey words: Parameter Identification, Lemaitre damageAbstract. A proper set of material parameters is one of the most important aspectsfor a successful simulation of metal forming processes. Several issues must be observedwhen choosing the constitutive relation and corresponding material parameters, amongstwhich the most important are: (i) the magnitude of the plastic deformation of the targetforming operation must be contemplated by the parameters of the constitutive model, (ii)possibility of failure prediction in fracture-free materials, and (iii) accurate prediction ofgeometrical changes caused by plastic deformation. Within this framework, the presentarticle discusses techniques to obtain constitutive parameters of a Lemaitre-type materialmodel. The strategy requires compliance of multiple tensile tests with specimens preparedaccording to different technical standards. Parameter identification is regarded as aninverse problem and solved using optimization methods.1 INTRODUCTIONSince the pioneering works of Tresca, Huber, von Mises and Hencky [1] on modellinginelastic deformation, many researchers have dedicated much effort to design constitu-tive formulations able to describe several types of materials, such as polymers, metalsand composites amongst many others. Notwithstanding, the determination of the cor-responding material parameters have constituted a real challenge. Therefore, for manyyears, identification of material parameters was based only on mechanical tests assuminguniform stress states. Furthermore, complex constitutive models involving large inelasticdeformations, non-uniform stress states and material degradation have posed new de-mands. In recent years, optimization techniques have been proposed to identificationproblems aiming at circumventing such difficulties. The present work is inserted withinthis framework in which an optimization method is used to parameter identification of aLemaitre-type material model based on tensile tests of specimens of different geometry.193M. Vaz Jr. and M. Tomiyama2 MATERIAL MODELLINGThe literature shows a growing number of material models able to account for me-chanical degradation. In general, such models, known as damaged materials, include newinternal variables and evolution laws which are fully coupled to the underlying plasticityproblem. The great advantage of damaged materials is the possibility to predict fail-ure onset in actual metal forming operations without pre-defined cracks. Availability ofdamage models in commercial codes has instigated further investigations on robust andefficient parameter identification strategies. The present work addresses a Lemaitre-typematerial model, as briefly described in the next session.2.1 Lemaitre-type damage modelA significant number of damage formulations is based upon thermodynamics of irre-versible processes and void growth concepts. Kachanov’s [2] and Rabotnov’s [3] earlyworks, and Lemaitre’s [4] equivalence principle laid foundations for further developments,giving rise to material descriptions oftentimes referred as Continuum Damage Mechanics(CDM). The present work uses an extension of Lemaitre’s damage model that accountsfor void closure and void opening effects. The reader is referred to [5, 6, 7] for furtherdiscussions on the damage formulation.The present constitutive model requires identification of hardening, ph, and damage,p l, parameters. Swift’s [8] hardening equation, σY = k(ε¯p + ε0)n is adopted in this work,where ε¯p is the equivalent plastic strain and ph = {k , ε0 , n} is the set of hardeningparameters. The damage parameters of the model are p l ={S , h+ , h− , εDp}. There-fore, the set of design variables for the present formulation can be generally expressed asp = ph ∪ p l.3 OPTIMIZATION-BASED PARAMETER IDENTIFICATIONParameter identification is a class of inverse problems which determines material prop-erties from a known response. The present problem is formulated using unconstrainedoptimization and accounts for experimental data obtained from tensile tests performedwith specimens of different geometry. Therefore, the first step of the optimization problemis formulated asMinimise g(p) =ns∑s=1λsgs(p) p ∈ RndSuch that pinfi ≤ pi ≤ psupi i = 1, . . . , nd, (1)where g(p) is the global objective function (global fitness), p = [p1 p2 · · · pi · · · pnd ]T isthe design vector containing nd material parameters pi, and psupi and pinfi are lateral con-straints. The global fitness, g(p), comprises contributions from ns individual mechanicaltests, so that λs is the weight function (∑nss=1 λs = 1), and gs(p) is the individual fitnessand represents a quadratic relative error measure between the experimental, RExps , and294M. Vaz Jr. and M. Tomiyamacorresponding computed forming load, R (p)Nums , of a mechanical test “s”,gs (p) =���� 1NsNs∑j=1(RExps,j −R (p)Nums,jRExps,j)2, (2)in which Ns is the number of experimental points of a mechanical test “s”. Particle SwarmOptimization [9, 10] is adopted to solve the optimization problem established in Eqs. (1)and (2) owing to the nature of damage models. Its suitability to this class of problemswas extensively discussed in Reference [11].It is important to note that solution of the optimization problem described in Eqs. (1)and (2) is associated with a given set of weights, λs. Inclusion of λs as design variablewould lead to a set of parameters which provide a minimum for one of the tests only.Therefore, the best set of weights must be determined by an additional optimizationproblem.This work proposes a new strategy by comparing the best individual fitness of eachtest, gmins , with their counterpart, gs, calculated evaluated by solution of Eqs. (1) and(2). Therefore, the new optimization problem is solved with weight functions λs as designvariables, so thatMinimise G(λ) λ ∈ RnsSuch that 0 ≤ λs ≤ 1 s = 1, . . . , nsns∑s=1λs = 1, (3)in which G(λ) is the mean global error,G(λ) =���� 1nsns∑s=1[gmins − gs(p)gmins]2, (4)which expresses a quadratic relative error between the best approximation, gmins , for eachmechanical test and individual fitness, gs(p), obtained by solving the optimization prob-lem established in Eq. (1). It is important to mention that there is a unique relationbetween the individual fitness and weight for each mechanical test, gs(p)⇔ λs.Numerical experiments show that the proposed optimization scheme represented byEqs. (3) and (4) is convex, making possible to use the Nelder-Mead optimization algorithm[12]. The optimization scheme establishes a polytope of ns + 1 vertices, upon whichexpansion/contraction/shrinkage operations are applied. As the optimization processadvances, the polytope moves towards the minimum and decreases in size. Therefore, themean relative distance between the centrepoint and all vertices of the polytope is used asconvergence measure,ϕ(k)d =d(λ(k))d(λ(0)), where d(λ(k)) =1ns + 1ns+1∑j=1���� ns∑s=1(λ(k)j,s − λ¯(k)s)2, (5)395M. Vaz Jr. and M. Tomiyamain which (k) indicates the iterative step, λ¯(k)s = 1/(ns + 1)∑ns+1i=1 λ(k)s,i is the coordinate ofthe centrepoint in the direction s = 1, · · · , ns, and ns + 1 is the number of vertices of thepolytope. In this work, convergence is assumed when the initial polytope reduces morethan 1,000 times its initial size, i.e. TOLd  10−3.4 NUMERICAL EXAMPLES AND DISCUSSIONSThe identification procedure was performed using tensile tests of cylindrical specimensprepared according to the American ASTM E 8M [13] and Brazilian ABNT NBR ISO6892 [14] technical standards. Extensometers with initial gauge length l0 = 25 mm or l0 =50 mm were used according to the specimen with maximum crosshead speed 3 mm/min.The specimens adopted in this work are referred as follows:ASTM #1: initial gauge length l0 = 25 mm and diameter d0 = 6.0 mm,ASTM #2: initial gauge length l0 = 50 mm and diameter d0 = 12.54 mm,NBR #3: initial gauge length l0 = 50 mm and diameter d0 = 10.0 mm,in which d0 is the initial diameter.The geometrical model considers axisymmetry around the rotation axis Z − Z ′ andsymmetry about the R − R′ axis, making possible to model only 1/4 of the specimen. Itwas adopted a structured, eight-noded quadrilateral finite element mesh with 200 elementsand 661 nodes with progressive refinement towards the specimen R−R′ axis. The meshesused for ASTM #1 e NBR #3 specimens were geometrically proportional to ASTM #2with identical element topology.Figure 1(a) shows the evolution of the convergence index, ϕd, whereas Figure 1(b)presents the corresponding evolution of the weights, λs, of the best vertex of the NMpolytope. It was assumed λ(0)s = 1/3 as initial values. After some variations in thebeginning of the iterative process, the optimization scheme quickly leads the weights tothe final values. Little change was observed after fourteen iterations and a convergencecriterion TOLd  10−2 could have been safely adopted (the maximum difference is 0.48% for λ1).The final material parameters, p, weights, λ, and mean global fitness, G(λ) are pre-sented in Table 1, whereas Figure 2 illustrates the corresponding load evolution. It isnoteworthy that the loading curves represent the best approximation to each individualtensile test with parameters determined simultaneously. The results show that geometrystill plays a role in determining of inelastic parameters of phenomenological descriptionsof plasticity phenomena. However, from industrial viewpoint, the results constitute agood approximation of complex phenomena as material degradation.5 FINAL REMARKSTensile tests have largely been used to determine inelastic parameters. The classicalprocedure accounts for plastic deformation up to necking onset in order to ensure uniformstress states. However, such condition is achieved for relatively small plastic strains.On the other hand, metal forming operations oftentimes involve large plastic strains.496M. Vaz Jr. and M. Tomiyama(a) Convergence index. (b) Weights, λs.   1.0E-041.0E-031.0E-021.0E-010 5 10 15 20 25 30StepConvergence index, dConvergence indexConvergence:  φ d < 10-3   0.280.30.320.340.360.380 5 10 15 20 25 30StepL3L2L1Weights, ConvergenceFigure 1: Evolution of the convergence index and weights.Table 1: Final material parameters, p, weights, λ, and mean global fitness, G(λ).Symbol ValueHardening k 1454.42 MPaParameters ε0 0.0461830ph n 0.414118Damage S 889.223 MPaParameters h− 0.00140679p l h+ 1.58511εDp 0.402607Weights λmin1 (ASTM #1) 0.31112λ λmin2 (ASTM #2) 0.35497λmin3 (NBR #3) 0.33390Global error G(λmin) 5.15307Aiming at such applications, this work uses inverse problem techniques to obtain materialparameters up to macroscopic failure. A Lemaitre-type damage model was utilised toassess material degradation e failure.In general, tensile tests of specimens of a single geometry is adopted. However, aimingat better prediction of the inelastic parameters, this work requires simultaneous compli-ance of tree tensile tests performed with specimens prepared according to two differenttechnical standards. The inverse problem technique used multi-objective optimization597M. Vaz Jr. and M. Tomiyama    0204060801000 0.2 0.4 0.6ElongationNumerical ExperimentalLoad [kN]ASTM #1 (d0 = 6 mm)ASTM #2 (d0 = 12.54 mm)NBR #3 (d0 = 10 mm)Figure 2: Load evolution for specimens ASTM #1, ASTM #2 and NBR #3.with the requirement of obtaining the best possible objective function for each individualspecimen. Bearing in mind the phenomenological character of the constitutive model, theresults indicate a good approximation for all tests. The large discrepancies were foundfor tensile tests involving the larger tensile loads since the individual objective functionhas been defined as a relative measure.AcknowledgementsThe author acknowledges the financial support provided by the Brazilian fundingagency CNPq - (National Council for Scientific and Technological Development), grantnumber 303412/2016-0.REFERENCES[1] Vaz Jr., M., de Souza Neto, E.A. and Mun˜oz-Rojas P.A. Materials modelling - chal-lenges and perspectives. In: M. Vaz Jr.; E.A. de Souza Neto; P.A. Muoz-Rojas.(Eds.), Advanced Computational Materials Modeling: From Classical to Multi-ScaleTechniques. Wiley-VCH, 1–22 (2011).[2] Kachanov, K.M. Time of the rupture process under creep conditions. Isv. Akad.Nauk. SSR. (1958) 8:21–31.[3] Rabotnov, Y.N. Creep Problems in Structural Members. North-Holland, (1969).[4] Lemaitre, J. A continuous damage mechanics model for ductile fracture. J. Eng.Mater. Technol., Trans. ASME (1985) 107:83–89.698M. Vaz Jr. and M. Tomiyama[5] de Souza Neto, E.A., Peric, D., Owen, D.R.J. Computational Methods for Plasticity.Theory and Applications. Wiley, (2008).[6] Vaz Jr., M., de Santi Jr., N., de Souza Neto, E.A. Numerical prediction of ductilefailure onset under tensile and compressive stress states. Int. J. Damage Mech. (2010)19:175–195.[7] Vaz Jr., M., Munoz-Rojas, P.A., and Lange, M.R. Damage evolution and thermalcoupled effects in inelastic solids. Int. J. Mech. Sci. (2011) 53:387–398.[8] Swift, H.W. Plastic instability under plane stress. J. Mech. Physics Solids (1952)1:1–18,[9] Eberhart, R., Kennedy, J. A new optimizer using particle swarm theory. In: Pro-ceedings of the 6th International Symposium on Micro Machine and Human Science.IEEE Press, 39–43 (1995).[10] Kennedy, J., Eberhart, R. Particle Swarm Optimization. In: Proceedings of the IEEEInternational Conference on Neural Networks. IEEE Press, 1942–1948 (1995).[11] Vaz Jr., M. Particle swarm optimization and identification of inelastic material pa-rameters. Eng. Comput. (2013) 30:936–960.[12] Nelder, J.A., Mead, R. A simplex method for function minimization, amage evolutionand thermal coupled effects in inelastic solids. The Comput. Journal (1965) 7:308–313.[13] ASTM - E 8M-86a. Tensile testing of metallic materials. ASTM, (2002).[14] ABNT NBR ISO 6892. Metallic materials - Tensile testing at ambient temperature(in Portuguese). ABNT, (2002).799

Further studies on identification of inelastic parameters for damaged materials

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Further studies on identification of inelastic parameters for damaged materials

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