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1. 2an pu expt Ps 3 where s is the plastic arc length o is the initial uniaxial yield stress and R and are the non linear isotropic hardening parameters For kinematic hardening the Armstrong Frederick law with the non linear kinematic hardening parameters H in and H is considered pe 2H sa 4 The hypo elastic material behaviour is described using Hooke s law relating the Cauchy stress o to the elastic strains through the Lam constants G and A 6 2GI5 AI 1 e9 C e 5 where C is the corresponding fourth order tensor Based on the constitutive equations above the stress update equations can now be formulated The state variables at the beginning of an increment are characterized by a superscript whereas those at the end of the increment are characterized by a superscript and the mid step configurations are characterized by a superscript In the finite strain case rigid body rotation is accounted for to preserve material objectivity henceforth a corotational formulation is used in order to transform all tensor valued state variables into the same rotation neutralized coordinate frame In this rotation neutralized coordinate frame the classical update scheme of the plastic arc length s of the plastic strain of the stress o and of the back stress is 1 Dan DUMITRIU Gaston RAUCHS Veturia CHIROIU 318 slas 248 6 apl _ ap 0 lt Eh e AgN2. 7 6 C A 2GAgN 8 oS ee ie H kin 1 exp H Ag 1 exp H Ag N 9 nl 1 2 with the total strain increment A calculated according to the midpoint rule and the normalized tensor N defined by 5 Pis 6 a _ P 6 4 m Pea A The plastic consistency parameter Ag is obtained from the equation expressing the yield function 1 for plastic material behaviour A 0 f P 6 2GP Ae H j 2GAg 1 exp H A8 K 0 11 nl A local Newton Raphson scheme is used to determine Ag from 11 3 UNIAXTAL INDENTATION TEST The uniaxial indentation test consists in pushing a hard indenter vertically into the plane surface of the material specimen Fig la A nanoindenter can record the load P and the penetration Ayp of the indenter tip into the surface where the penetration Aup is the total displacement of the specimen contact surface at the vertical line of symmetry The spherical tip of the indenter has hypo elastic material behaviour with Young s modulus E 1016 GPa and Poisson s ratio v 0 07 indenter indenter tip specimen ee am PE ae H fh A Fig 1 a Scheme of the uniaxial indentation test b Axisymmetric finite element model of the indentation test 1 material specimen a b 319 Optimization Procedure for Parameter Identification in Inelastic Material Indentation Testing In this inelastic
3. where simulated and pseudo experimental residual imprints are compared u r the simulated total vertical displacement at the fixed radial location r u r simulated total vertical displacement at the reference point r e g the imprint centre u r pseudo experimental total vertical displacement at the fixed radial location r u r pseudo experimental total vertical displacement at the reference point r The gradient of the objective function with respect to the variables vector x is V 1 a a e Oe OF OF OF oe is x Oke x x x x OG E v o R OB OH OH where the derivative of the objective function with respect to the material parameter x i 1 7 follows from 14 O y sim k exp k Ohi ce a LP Hig PD 16 sim y _ sim p0 42 lus r ysim r ueP r uP r ous r 2 u r i z X i sim k i i Pha PN ng Quen u Ox Ox state variables with respect to the material parameter x The direct differentiation method is used to perform this sensitivity analysis as described in 1 The elements Hj of the Hessian matrix H i e the second order ilei er aye derivatives can be deduced by deriving 16 OX OXx As for the derivatives they are computed from the derivatives of the 321 Optimization Procedure for Parameter Identification in
4. elasto plastic strains called SPPRc is used This software implements the stress update equations and models the contact between the spherical tip indenter and the unknown inelastic material 1 The stress update equations are formulated based on the constitutive equations for hypo elastic material behaviour and considering associative plasticity with isotropic J2 flow theory Using SPPRc software and initializing the seven unknown parameters with initial guesses in the range between some lower and upper bounds an optimization procedure is launched in order to minimize the objective function i e the difference or gap between the experimental load penetration curve and the simulated one So by means of the optimization procedure the material parameters are updated until the minimization of the objective function is achieved Several approaches are available for this optimization procedure from classical gradient based numerical optimization to modern neural networks approach Gradient based numerical optimization methods 5 were preferred here knowing that neural networks need a big amount of 317 Optimization Procedure for Parameter Identification in Inelastic Material Indentation Testing objective function evaluations and here the computational time for one evaluation is not negligible Ponthot and Kleinermann 2 proposed to solve parameter identification inverse problems using a cascade optimization methodology i e a robust and efficient
5. virtual initial guesses the optimization procedure proposed in 4 is supposed to converge to the material parameters used to obtain the pseudo experimental data i e the ones in Table 1 Two such convergence tests T1 and T2 were performed Table 3 presenting the virtual initial guesses used in these two tests as well as the lower and upper bounds of the seven parameters to be identified plus the solution of the problem i e the pseudo experimental material parameters from Table 1 Table 3 Different values for the material parameters T1 and T2 initial guesses upper and lower bounds problem s solution E MPa v o MPa R MPa BIJ iin MPa Ani T1 initial guesses 100000 0 2 800 50 200 50 100 T2 initial guesses 261110 0 34 667 46 523 398 74 Artificial experiment 200000 0 3 400 100 100 200 50 Lower bounds 30000 0 05 1 0 10 0 10 Upper bounds 600000 0 49 1000 1000 1500 1000 1500 The proposed optimization procedure consists in a sequence of fours stages e Gauss Newton method 18 applied on variables E and o the other variables being blocked on their initial guesses called G N1 maximum 8 iterations allowed e Gauss Newton method 18 applied on all seven variables called G N2 maximum 10 iterations e Levenberg Marquardt method 19 applied on three variables the variables located temporarily on the bounds then the variables x corresponding
6. 000 7000 D 4 4 6000 a e A foo e 5000 gt 4000 M roS eee eeee seen ees s _ 3000 2000 4 o M Ae 1000 Loe seccsesesescseseseseses oe 9 10 0 0 05 0 1 0 15 penetration into surface microns load microN 0 25 Fig 2 Artificial pseudo experimental load penetration curve for single load unload cycle Dan DUMITRIU Gaston RAUCHS Veturia CHIROIU 320 4 PARAMETER IDENTIFICATION OPTIMIZATION PROCEDURE Let x denote the vector of the seven material parameters to be identified x E v o R B Hi Hy 13 The material parameter identification is performed using an optimization procedure i e one has to minimize bring to zero if possible the difference between the simulated curve and the pseudo experimental one Both load penetration curve and residual imprint mapping data are considered for comparing simulation with pseudo experimental results The objective function to be minimized is thus N N min Py ner PO Y fee i E U 14 k l l l with N the number of points in which simulated and pseudo experimental penetrations are compared h P the simulated penetration of the indenter tip into the surface corresponding to the load P at time using the material parameters computed at that stage of the optimization procedure _ hj P pseudo experimental penetration corresponding to load P N number of fixed radial locations r
7. Inelastic Material Indentation Testing A D tip tip ong PE ohnsim PK 8h sim PK RA PS za Faget nee Ps es h MORI Ox ax ey J lauren r ysim r ousi r ysim o l 1 Lem r ysim r ue r uP r o us r ysim r l Ox OX The optimization procedure proposed in this paper is a sequence of two gradient based optimization methods i e the Gauss Newton method and the Levenberg Marquardt method The optimization problem here is a minimization with simple bounds i e the variables of the minimization problem are limited by constant lower and upper bounds The Gauss Newton optimization method is an iterative method based on the following recurrence relation 4 5 xP x pP B VE 18 where p denotes here the iteration number and u is the parameter of the line search algorithm performed inside the Gauss Newton method to find the lowest value of the objective function along the search direction H VE The Hessian matrix H given by 17 has to be positive definite in order to provide that the search direction is a descent direction 3 i e in order to converge towards a minimum of the objective function As for the Levenberg Marquardt optimization method 3 5 its recurrence relation is x xP JP T J EPI Jr TE 19 where J is the Jacobian defined here as _ OF CE OB OB om vz 20 E
8. SISOM 2007 and Homagial Session of the Commission of Acoustics Bucharest 29 31 May OPTIMIZATION PROCEDURE FOR PARAMETER IDENTIFICATION IN INELASTIC MATERIAL INDENTATION TESTING Dan DUMITRIU Gaston RAUCHS Veturia CHIROIU Institute of Solid Mechanics Romanian Academy Bucharest Romania Laboratoire de Technologies Industrielles Centre de Recherche Public Henri Tudor Grand Duchy of Luxembourg Corresponding author D Dumitriu Address Str Ctin Mille nr 15 Sector 1 010141 Bucharest Romania Fax 40 21 3126736 Email dumitri04 yahoo com The indentation of an inelastic material isotropic half space with a spherical tip indenter is simulated using an in house finite element code written in Fortran In order to identify seven unknown material parameters E and v for elasticity o R B Akin and Hu for plasticity the simulation results must match as good as possible the experimental load penetration curve obtained when indenting the inelastic material This material parameter identification is performed using an optimization procedure i e one has to minimize the difference between the experimental curve and the simulated one The optimization procedure used here starts by applying a Gauss Newton method then continues using a Levenberg Marquardt method Using this optimization procedure the material parameter identification was successfully achieved for the considered indentation case study Keywords Indentation Param
9. broutines for Mathematical Applications Visual Numerics Inc pp 867 1029 Chapter 8 Optimization 1997 5 NOCEDAL J WRIGHT S J Numerical Optimization Springer Verlag New York 1999
10. ction G N1 G N2 L M1 L M2 gt lt gt lt gt lt Sbe amp eNHSdGbkBHRAO logyo objective function N P gt b second test T2 a first test Tl Fig 3 Reduction of log with the number of iterations for a first test T1 b second test T2 6 CONCLUSIONS The framework of this work is the parameter identification related to inelastic material indentation testing The goal of this inverse problem is to determine the unknown material parameters of a specimen in order to reach some experimental data measured on that specimen More precisely the purpose of this paper was to propose a reliable variant of optimization procedure for this material parameter identification problem Based on IMSL Fortran Math Library 4 subroutines the optimization procedure was implemented in SPPRc an in house FEM software for finite elasto plastic strains The optimization procedure was divided in four stages two stages based on Gauss Newton method followed by two stages based on Levenberg Marquardt method The convergence results towards the desired global minimum were satisfactory Further investigations including additional load cycles of the indentation test are necessary in order to increase and guarantee the reliability of this parameter identification optimization procedure Gradient based optimization methods such as Gauss Newton and Levenberg Marquardt methods are local optimizers bearing the nec
11. essity to supply an appropriate initial guess in order to place the optimization problem closer to its searched global optimum So further work may provide a more elaborated initialization of the unknown material parameters not just a simple guess ACKNOWLEDGEMENTS Gaston RAUCHS gratefully acknowledges the financial support of the Fonds National de la Recherche FNR of the Grand Duchy of Luxembourg through grant number FNR 05 02 01 VEIANEN Dan DUMITRIU and Veturia CHIROIU gratefully acknowledge the financial support of the National Authority for Scientific Research Council ANCS Romania through CEEX postdoctoral grant nr 1531 2006 which funded the visit of Dan DUMITRIU at the Centre de Recherche Public Henri Tudor REFERENCES 1 RAUCHS G Optimization based material parameter identification in indentation testing for finite strain elasto plasticity ZAMM Z Angew Math Mech 86 7 pp 539 562 2006 2 PONTHOT J P KLEINERMANN J P A cascade optimization methodology for automatic parameter identification and shape process optimization in metal forming simulation Comput Methods Appl Mech Engrg 195 pp 5472 5508 2006 3 SEIFERT T SCHENK T SCHMIDT I Efficient and modular algorithms in modeling finite inelastic deformations Objective integration parameter identification and sub stepping techniques Comput Methods Appl Mech Engrg 196 pp 2269 2283 2007 4 IMSL Math Library vol 1 and 2 FORTRAN Su
12. eter Identification Optimization Procedure Gauss Newton Levenberg Marquardt convergence 1 INTRODUCTION This paper deals with the parameter identification in indentation testing of inelastic materials It is an inverse engineering problem very relevant to industry the purpose being to determine one or more inelastic material parameters that would lead to the most accurate agreement between experimental data and indentation simulation results Since indentation modelling involves non uniform stress fields to be analyzed and non linear material behaviour there is no straight forward analytical solution for material parameter identification associated to indentation testing So numerical approaches finite element modelling in this case must be used for obtaining the fields of the state variables corresponding to the indentation test Suppose one has to determine the unknown parameters of a new inelastic material Using a nanoindenter experimental load penetration curves are available for the new unknown material Based on this experimental data the purpose is to identify the unknown material parameters Young s modulus E and Poisson s ratio v elasticity parameters respectively the initial uniaxial yield stress o the non linear isotropic hardening parameters R and B and the non linear kinematic hardening parameters H in si Hm plasticity parameters In what concerns the simulation of the indentation an in house FEM code for finite
13. material indentation problem the parameter identification concerns seven unknown material parameters Young s modulus E and Poisson s ratio v elasticity parameters respectively the initial uniaxial yield stress o the non linear isotropic hardening parameters R and and the non linear kinematic hardening parameters Hii Si Hy plasticity parameters In fact the linear hardening parameters R and Akin have been replaced with the following parameters which both have the physical meaning of a stress Hi Hn md R 12 H nl B The axisymmetric finite element model for spherical indenter tip and material specimen is shown in Fig 1b where 112 bi quadratic elements were used to model half of the material specimen and 47 bi quadratic elements were used to model half of the spherical tip indenter The simulations are performed using SPPRc which implements the stress update equations 6 9 for this axisymmetric finite element model considering finite elasto plastic strains In SPPRc the contact modelling between the indenter tip and the material specimen is realized by inhibiting the geometrical interpenetration of the two bodies This can be achieved by applying tractions over the contact area The surface of the indenter is the master surface whereas the nodes on the material specimen surface are the slave nodes In simple contact formulations like penalty or augmented logarithmic barrier methods the tractions to be applied on the contac
14. sequence of gradient based optimization methods The optimization procedure proposed here is a sequence of two gradient based optimization methods i e the Gauss Newton method and the Levenberg Marquardt method The gradient of the objective function i e the vector of the objective function derivatives with respect to the unknown parameters is computed in a sensitivity analysis using the direct differentiation method to calculate the derivatives of the state variables with respect to the material parameters This direct differentiation method calculates directly the derivatives through an incremental linear update scheme during the incremental solution of the direct deformation problem 1 The computational effort is thus considerably reduced compared with the numerical calculation of the derivatives by finite differences 2 THE MATERIAL BEHAVIOUR CONSTITUTIVE AND STRESS UPDATE EQUATIONS Associative plasticity with isotropic J2 flow theory is considered 1 Plastic yielding is governed by the yield function f f lt 0 for elastic material behavior f P o a K f 0 for plastic material behavior 1 f gt 0 excluded where K is the yield limit the back stress an internal variable o is the Cauchy stress and P is the deviatoric projection operator P 1 i1 1 2 where I is the fourth order symmetric identity tensor The variation of the yield limit K is described through isotropic hardening with 1 K
15. t boundary for inhibiting interpenetration are calculated from the local gap between slave nodes and their projection on the master segment More details about the contact modelling can be found in 1 The indentation test considered here is in fact a virtual indentation test the purpose being mainly to validate the proposed material parameter identification optimization procedure More precisely a simulation of the indentation test is carried out with known material parameters and the resulting load penetration curve see Fig 2 is used as artificial experimental data called also pseudo experimental data The material parameters used to obtain the pseudo experimental data are given in Table 1 The parameter identification optimization procedure is then started with a different set of material parameters a virtual initial guess and should ideally converge to the material parameters used to obtain the pseudo experimental data Table 1 Material parameters used in the artificial experiment E MPa v o MPa R MPa p H Hun MPa Hu Artificial experiment 200000 0 3 400 100 100 200 50 The artificial pseudo experimental load penetration curve considered in this paper concerns only a single load unload cycle see Fig 2 Of course more complex cycles can be considered e g load unload reload load unload cycle also residual imprint mapping data can be taken into consideration 8
16. to the smallest oe the other four variables being Ix l blocked on their previous values called L M1 maximum 15 iterations allowed e Levenberg Marquardt method 19 applied on all seven variables called L M2 maximum 30 iterations allowed Fig 3a shows the convergence of the optimization procedure for the first numerical test T1 where the objective function is minimized up to 1 28 10 ym Fig 3b shows the convergence results for min T1 the second numerical test T2 where the objective function is minimized up to mintz 7 16 10 um Both figures show the reduction of log with the number of iterations performed The vertical dotted lines indicate the switches between the four stages of the optimization procedure The results are satisfactory showing the reliability of the proposed four stages optimization procedure based on Gauss Newton and Levenberg Marquardt methods The objective function is always decreasing for both tests presented without diverging or blocking in undesired local minima The objective function has been practically brought to zero the material parameters obtained by this identification procedure matching almost perfectly the considered pseudo experimental material parameters 323 Optimization Procedure for Parameter Identification in Inelastic Material Indentation Testing number of iterations number of iterations 0 5 10 15 20 25 30 35 40 45 eaYNdbdbbk wR OO logy objective fun
17. v o OR OB OH OH ae and where the scalar is determined as follows Er 10 DO amp 10 amp 21 UNTIL matrix J J I becomes positive definite The implementation in SPPRc software of the Gauss Newton and Levenberg Marquardt methods was done by means of the respective subroutines provided by IMSL Fortran Math Library 4 5 CONVERGENCE RESULTS The parameter identification optimization procedure proposed in 4 was applied to the virtual uniaxial indentation problem described in 3 The pseudo experimental load penetration curve is the one shown previously in Fig 2 where only a single load unload cycle has been considered The residual imprint mapping data taken into account in provided in Table 2 with the reference point located on the vertical symmetry axis i e r 0 Dan DUMITRIU Gaston RAUCHS Veturia CHIROIU 322 Table 2 Residual imprint mapping data index 1 2 3 4 5 6 7 8 9 r um 0 125 0 250 0 375 0 500 0 625 0 750 0 875 1 000 1 125 u P r u r um 0 0035 0 0127 0 0279 0 0495 0 0775 0 1121 0 1536 0 2019 0 2462 10 11 12 13 14 15 16 17 18 19 20 1 250 1 375 1 500 1 625 1 750 1 875 2 000 2 125 2 250 2 375 2 500 0 2383 0 2314 0 2265 0 2231 0 2205 0 2185 0 2168 0 2156 0 2146 0 2138 0 2132 Starting from various
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