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Journal of Materials Processing Technology 75 (1998) 204–211

Determination of the elasto-plastic properties of aluminium usinga mixed numerical–experimental method

M.H.H. Meuwissen, C.W.J. Oomens *, F.P.T. Baaijens, R. Petterson, J.D. JanssenDepartment of Mechanical Engineering, Eindho6en Uni6ersity of Technology, PO Box 513, 5600 MB Eindho6en, The Netherlands

Received 13 January 1997

Abstract

A mixed numerical–experimental method is used to determine the parameters in elasto-plastic constitutive models. Aluminiumplates of non-standard geometry are mounted in a uniaxial tensile-testing machine. On the surfaces of the plates retro-reflectivemarkers are placed. The displacements of these markers are measured optically. These measurements are used to determine yieldstresses in the isotropic Von Mises and the orthotropic Hill yield criterion. The models are evaluated by examining the estimatedparameters and the residual displacements. © 1998 Elsevier Science S.A.

Keywords: Parameter estimation; Aluminium; Yield criteria; Finite-element modelling; Inverse methods

1. Introduction

To characterise the mechanical behaviour of a mate-rial quantitatively, a constitutive model must be chosenand the parameters in this model must be determinedfrom measurements. In standard methods, the determi-nation of parameters is based on the use of test sampleswith a well defined standardised geometry and loading,such that particular conditions on the stress and strainfield are satisfied in (a part of) the sample. Then, theunknown model parameters are obtained via analyticalrelationships from measurements of, for example,forces, torques, displacements and/or twist angles.

In some situations, the application of standard meth-ods is inconvenient or even impossible. For example,for many biological and composite materials, takingout a test sample from a larger structure can be unac-ceptable, because this disrupts the internal coherence ofthe material. Moreover, if the material is inhomoge-neous or anisotropic, it is generally impossible to satisfythe conditions on the stress and strain field in any partof the sample. Therefore, an increasing interest is foundin the literature in so-called mixed numerical–experi-mental methods based on the use of non-standard

experiments in which the conditions on stress and strainfields are relaxed [1–9].

In many metal-forming processes, the materials areloaded at high strains, high strain rates and high tem-peratures. Of course, the reliability of constitutive mod-els depends strongly on the ability to measuremechanical properties in the range of strains, strainrates and temperatures that cover the whole process.Standard methods in this area are uniaxial tensile andcompression tests, torsion tests and plane-strain com-pression tests. If these tests are to be carried out in adomain comparable to that of the actual forming pro-cess (i.e. high strains, high strain rates and high temper-atures), they fail due to disturbances such as necking,stress localisation, frictional effects, or deformationheating in the material. For these reasons, a number ofresearchers have recently applied mixed numerical–ex-perimental methods [10–14] with these materials. Inmany of these applications, the sample geometry and/orloading is still very close to those of standard methods.Moreover, the experimental information is limited toquantities measured at the boundaries of the samples.

The mixed numerical–experimental method as pro-posed by Hendriks [3] is used here to determine theparameters in elasto-plastic models. In this method thegeometry and/or loading of samples is non-standard. Inaddition, next to boundary information, field informa-

* Corresponding author. Fax.: +31 40 2447355; e-mail:[email protected]

0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S 0 924 -0136 (97 )00366 -X

M.H.H. Meuwissen et al. / Journal of Materials Processing Technology 75 (1998) 204–211 205

Fig. 1. Block diagram of the mixed numerical–experimental method.

yield criterion. Finally, the results of estimation areevaluated.

2. The mixed numerical–experimental method

Fig. 1 shows a block diagram of the mixed numeri-cal–experimental method. In this method, three com-ponents can be discerned: (i) one or more experimentsin which non-uniform displacement fields are measuredon specimens that are more or less arbitrarily shapedand possibly multi-axially loaded; (ii) a numerical(finite-element) analysis of the experiment(s) in whichan assumed constitutive model of the material be-haviour is incorporated; and (iii) an estimation al-gorithm that determines estimates of the unknownparameters iteratively, based on an initial guess of theparameters and the difference between the measuredand calculated displacement fields.

In Fig. 1, u denotes the input (prescribed displace-ments, loads, etc) to the experiment and the analysis, mthe measured field information, h the output of thefinite-element analysis, u the a priori unknown parame-ters, u0 the initial guess of the parameters and H is thesensitivity matrix, i.e. H=(h/(u.

New estimates of the unknown parameters are deter-mined by minimising the following sum of squares:

J= (m−h(u))TV(m−h(u))+ (u0−u)TW(u0−u) (1)

where V and W are two positive definite symmetricweighting matrices. This sum of squares is minimisediteratively using the Gauss-Newton scheme. Iteration iscontinued until one of the following criteria is satisfied:the sum of squared residual displacements is smallerthan a critical value or the sum of squared parametersadjustments is smaller than a critical value. More de-tails about the algorithm can be found in [3].

tion is measured over the surface of the sample. Thisextra information is expected to be valuable in parame-ter determination and model validation, especially forinhomogeneous and/or anisotropic material behaviour.

The current investigation is a pilot for the use of themixed numerical–experimental method as proposed byHendriks for parameter determination at high strainsand high strain rates. For these initial studies theexperiments are still quasi-static and the strain levelsare low. However, the geometry of the sample is non-standard. Thin plates of commercially-available alu-minium A1200 are mounted in a standard uniaxialtensile-testing machine. Retro-reflective markers areplaced on the surface of the plates. The samples areloaded and the displacements of the markers are mea-sured optically. The measured displacement fields areused to determine yield stresses in the isotropicVon Mises yield criterion, and the orthotropic Hill [15]

Fig. 2. Definition of a, the angle from the global x-direction to the rolling direction (j) of the plate, characteristic dimensions of the samples andpositions of the retro-reflective markers on both samples.

M.H.H. Meuwissen et al. / Journal of Materials Processing Technology 75 (1998) 204–211206

Fig. 3. Displacement of the upper clamp versus measured clamp force for sample A (left) and sample B (right). The points marked as ‘*’ denotesteps at which the position of the retro-reflective markers are measured. Solid arrows mark the steps at which measurements are used to determinethe yield stresses in the constitutive models. Outlined arrows mark the steps used to determine parameters in the Nadai hardening law.

3. Experimental set-up and observations

The material investigated is a commercially-availablealuminium alloy 1200. From a 0.5 mm thick plate, twosamples were machined with dimensions 140×60 mm(Fig. 2). The rolling direction of the material is denotedby j, the transverse direction in the plane of the plateby h, and the direction perpendicular to the plate by z.The angle from the global x-axis to the rolling directionj is 0° for sample A and 60° for sample B (Fig. 2).

A total of 120 retro-reflective markers (0.7 mm di-ameter) is placed on the surface of each sample (Fig. 2).Before placing these markers, a thin coat of matt blackpaint is sprayed on to the surface, the coat preventingdisturbing reflections from the bare metal.

The samples are mounted ‘as-received’ in a standarduniaxial tensile-testing machine. The distance betweenthe clamps is approximately 100 mm. The samples areloaded by moving the upper clamp upwards in equal

steps of 2.5×10−2 mm from under displacement con-trol. The velocity of this clamp is 1.67×10−3 mm s−1.The lower clamp is fixed. The experiments are carriedout at room temperature. The prescribed clamp dis-placement versus measured clamp force is shown in Fig.3.

In the unloaded situation and at the end of eachloading step, the positions of the retro-reflective mark-ers are measured optically using a random access videotracking system (Video Interface 84.330, HentschellGmbH Hannover). Fig. 4 shows the measured displace-ments of the markers at the steps indicated by solidarrows in Fig. 3. The average magnitude of thesedisplacements is approximately 0.2 mm for the markersnear to the upper clamp and less than 0.1 mm for thosenear to the lower clamp. The standard deviation of themeasurement noise is approximately 3×10−3 mm. Themaximal principal strains are approximately 0.5% in theregion between the indentations. The inhomogeneity inthe displacement- and strain-field is caused mainly bythe geometry of the specimen and by non-uniformloading due to slip in the clamps.

4. Set-up of estimation process

4.1. Finite-element model and boundary conditions

Finite-element models of the samples are imple-mented in the package MARC [16]. The meshes ofsample A and sample B consist of 1782 and 1751elements respectively. For both meshes, 4-noded iso-parametric plane-stress elements are used.

The positions of the markers near to the clamps areused to define the upper and lower borders of themodels. At the lower borders, the measured x- andy-displacements of the markers are used as kinematicboundary conditions in the models. At the upper bor-ders, the same is done with the measured x-displace-

Fig. 4. Measured displacements of the retro-reflective markers (×50)at the steps marked with solid arrows in Fig. 3 for sample A (left) andsample B (right).


Fig. 5. Estimated Von Mises yield sv0 stress versus iteration count (left) for measurement data of sample A (s60=81.0 N mm−2), sample B(s60=69.6 N mm−2) and combined (s60=76.2 N mm−2) and comparison between the predictions of the models and the results of the uniaxialtensile-test (right).

Fig. 6. Residual displacements (×350) for the Von Mises yield criterion for sample A (left) and sample B (right), with combined use ofmeasurement data.

ments. Furthermore, it is assumed that all nodes atthese borders have equal y-displacements. The actualy-displacements of the top markers vary a few percentdue to the inhomogeneous deformation in the sampleand slip in the clamps. The measured clamp force isapplied as a prescribed y-force on these nodes. Fornodes at the upper and lower borders that do notcoincide with markers, the prescribed displacements arelinearly interpolated between the two nearest bordernodes that do coincide with markers. The boundaryconditions are applied in 30 equal increments.

4.2. Constituti6e models

To describe the mechanical behaviour of the alu-minium, elasto-plastic constitutive models are em-ployed. In the elastic region, the material is assumed tobehave isotropically (Hooke’s law). This behaviour isconfirmed in standard uniaxial tensile-tests on thismetal. Due to previous inelastic deformations duringthe forming process of the plate, the aluminium willexhibit anisotropic plastic behaviour. Depending on thedeformation history, the anisotropy is of different type.For rolled plate or sheet metal, plastic orthotropy isusually assumed: two axes of symmetry lie in the plane

of the plate (j and h, Fig. 2) and the third axis (z) isperpendicular to the plate.

For describing plastic behaviour, two different mod-els are applied. Although this behaviour is expected tobe anisotropic, the first model incorporates theisotropic Von Mises yield criterion, whilst the secondmodel incorporates the Hill [15] yield criterion. Forplane-stress situations the latter criterion states thatplastic yielding occurs if the following condition is met:

Fs2jj+Gs2

hh+H(sjj−shh)2+2Ns2jh=1 (2)

where sjj, shh, and sjh are the components of the stresstensor in the j-h-coordinate frame (Fig. 2) and F, G,H, and N are parameters given by:

2F= (Y−2+Z−2−X−2)

2G= (Z−2+X−2−Y−2)

2H= (X−2+Y−2−Z−2)

2N=T−2 (3)

Here, X, Y and Z are the yield stresses in the j-, h- andz-direction of the material, respectively and T is theshear yield stress in the jh-plane. Note that for X=Y=Z and T= (1/3) 3 X this criterion reduces to theVon Mises yield criterion. Furthermore, the associative


Table 1Estimated standard deviation of the residual displacement components (s)

Hardening law Measurement data Estimated standard deviation s (mm)Yield criterion

Linear Sample A 6.05×10−3Von Mises4.98×10−3Sample BVon Mises Linear6.83×10−3CombinedVon Mises Linear

Linear CombinedHill 5.51×10−3

Sample A 4.75×10−3NadaiVon Mises

Standard deviation of measurement noise: s:3×10−3 mm

plastic flow is assumed. For both models, linearisotropic work hardening is chosen:

s= s0+Lop (4)

where s is the current yield stress, s0 is the initial yieldstress, L is the plastic modulus and op is the equivalentplastic strain.

4.3. Parameters u, measurements m and weightingmatrices

The resolution of the measurement system was nothigh enough to measure the small elastic strains suffi-ciently accurately. Therefore, Young’s modulus E andPoisson’s ratio n were taken from uniaxial tensile-tests(E=7.0×104 N mm−2, n=0.31). In addition, thevalue of the plastic modulus was taken from the sametests (L=1.0×104 N mm−2). In the first model (VonMises yield criterion) the following parameter set willbe estimated:

uT1 = [sv0] (5)

where sv0 is the initial Von Mises yield stress. Theinitial estimate of this parameter is:

uT10= [70.0] (6)

In the second model (Hill yield criterion) the parameterset is given by:

uT2 = [X0, Y0, Z0, T0] (7)

where X0, Y0, Z0 and T0 are the initial yield stresses.The initial estimates are:

uT20=

�70.0, 70.0, 70.0,

70.0

3

nNote that for these values, the Hill criterion equals theVon Mises criterion with sv0=70.0 N mm−2.

The weighting matrix W is chosen diagonally forboth models. For the first model, the diagonal compo-nents are set to:

W1=Æ10−3É (9)

and for the second model:

W2=Æ10−3, 10−3, 10−3, 10−3É (10)

The diagonal components are approximately the in-verse of the squared expected error in the initial guessof the parameters.

The measurements column m contains the measuredx- and y-displacements of the markers in the regionbetween the two notches at the steps indicated withsolid arrows in Fig. 3. The weighting matrix V is chosendiagonally. The diagonal elements are set to the inverseof the variance of the measurement noise of the mea-surement system (Vii:105 mm−2). For each model,three estimation runs are carried out with different setsof measurement data: (1) measurement data of sampleA; (2) measurement data of sample B; (3) measurementdata of both samples.

5. Estimation results

5.1. Von Mises yield criterion

Fig. 5 shows the estimated Von Mises yield stress sv0

versus iteration count for the three estimation runs. Theestimates converge to stable values within two itera-tions. Several estimation runs were performed startingwith different initial parameter guesses. In most casesconvergence was established to the same final estimates.In some cases (very bad initial estimates), the parame-ters did not converge to stable values.

In Fig. 5 a comparison is made between the constitu-tive model, based on the estimated yield stress and twouniaxial tensile-tests in the material’s j- and h-direc-tion. Differences between the model-based predictionand the results of the uniaxial tensile-tests can be seenhere. The estimates of the initial yield stress are quitereasonable. However, the model predicts a lower yieldcurve than that measured in the uniaxial tensile-tests.Moreover, the estimated parameters depend on the setof measurements used.

The quality of the first model can be validated byexamining the residuals, i.e. the difference between themeasured and calculated displacements. If the modelgives an accurate description, the residuals are causedmainly by measurement noise. Thus, if the characteris-tics of the residuals differ from those of the measure-ment noise, the model has one or more defect(s). Here,


Fig. 7. Estimated yield stresses in the Hill criterion versus iteration count for the use of the measurement data of both samples (left) andcomparison of model versus uniaxial tensile-test data (right).

Fig. 8. Residual displacements (×350) for the Hill yield criterion for sample A (left) and sample B (right).

the noise is expected to be uncorrelated with a standarddeviation of s=3×10−3 mm. The residuals for thefinal estimates of the parameters are plotted in Fig. 6for the combined use of the measurement data. Cleardeterministic patterns are present in the residuals,which indicates the presence of systematic errors. Thestandard deviation s of the residuals is estimated from:

s=' 1

No−Np

(m−h(u. ))T(m−h(u. )) (11)

where No is the total number of observations (length ofthe column m), and Np is the number of estimatedparameters. It is assumed that the standard deviation isthe same for both of the residual fields. Table 1 sum-marises the estimated standard deviations of the resid-ual fields for the three estimation runs. It can be seenthat the estimated standard deviation of the residualdisplacements is approximately twice the standard devi-ation of the measurement noise. The estimated stan-dard deviation is about 5–6% of the magnitude of themeasured displacements. Hence, the overall fit of thesemodels on the experimental data is good.

5.2. Hill yield criterion

Based on the above observations, it can be concludedthat the first model has one or more (small) defects. An

obvious extension is the incorporation of the or-thotropic Hill criterion. The estimated parameters ofthis criterion versus iteration step are shown in Fig. 7for the combined use of the measurement data. Forestimation runs that use the measurement data for onlyone sample, convergence could not be established forall parameter estimates within 15 iterations. Appar-ently, the individual experiments are not suitable fordetermining all parameters in this criterion accurately.Convergence of the parameter estimates is establishedafter eight iteration steps. A comparison betweenmodel-based predictions and uniaxial tensile-test data isshown in Fig. 7. For both model and experiment theyield curve in the j-direction is lower than that in theh-direction. Nevertheless, the predicted yield curves arestill too low.

Fig. 8 shows the residual displacement fields for theconverged parameter estimates in the Hill criterion. Theestimated standard deviation s of these fields is slightlysmaller than that for the comparable run of the VonMises yield criterion (Table 1).

5.3. Nadai hardening law

A weakness of the previous models is the linear-hard-ening law given by Eq. (4). This law is replaced by thenon-linear Nadai hardening law:


Fig. 9. Estimated parameters in the Nadai hardening law (Eq. (12)): c (left) and n (right).

sv=sv0+conp (12)

where c and n are material constants. This law isimplemented in the model incorporating the Von Misesyield criterion. For estimating the parameters in thismodel, the measurement data of sample A, marked byoutlined arrows in Fig. 3, are used. Simultaneous esti-mation of the parameters sv0, c and n did not lead toconvergence. Therefore sv0 is fixed to 70 N mm−2 andc and n are estimated from the measurements [17]. Theestimated parameters versus iteration step are shown inFig. 9.

The residual displacement fields (not shown) arecomparable to those of the previous estimation runswith the linear-hardening law. The estimated standarddeviation is somewhat smaller than for the model incor-porating the linear-hardening law (Table 1).

Fig. 10 shows a comparison of the measured uniaxialtensile-test data and the prediction of the estimatedNadai hardening law. There is a good agreement be-tween the slopes of the curves. The model curve liesbelow the experimental data. Repeating the estimationruns with different values of sv0 does not lead to betterresults.

6. Discussion

A mixed numerical–experimental method is used todetermine the parameters in the isotropic Von Misesand orthotropic Hill yield criterion from displacementmeasurements on two samples showing inhomogeneousstress- and strain-distributions.

For Von Mises yield criterion in combination withthe linear-hardening law, the differences betweenmodel-based predictions and experimental data fromuniaxial tensile-tests were approximately 10–20%.Changing the set of measurement data used forparameter estimation resulted in different final esti-mates of the yield stress. Furthermore, deterministicpatterns could be discerned in the residual displacementfields. These observations indicated the presence ofunmodelled phenomena. The Hill criterion resulted in aslightly better fit of the experimental displacementfields, which could be expected since the algorithm hasmore freedom in fitting the model on the experimentsdue to the greater number of parameters. In addition,the incorporation of the non-linear Nadai hardeninglaw led to a better fit of the experimental data and acloser agreement with the predictions of the estimatedmodel and measurements obtained from uniaxial ten-sile-tests.

For all models, the overall fit of the model predic-tions on the experimental displacement fields is good(within a few percent). Nevertheless, even for the Hillyield criterion, deterministic patterns are visible in theresidual displacements. Hence, one or more (small)modelling errors are present. These could be caused bythe inherent inadequacies of the constitutive model. TheHill yield criterion is known to be rather inaccurate foraluminium [18]. Most certainly, the patterns in theresiduals are caused by the limited resolution of themeasurement system. Hence, better agreement betweenestimation results and the results of the uniaxial tensile-test is expected when a measurement system is used thatis better suited for the strain domain in the experiments,for example Moire interferometry or electronic patterninterferometry [19].

Fig. 10. Comparison between uniaxial tensile-test data and the Nadaihardening law.


The use of the mixed numerical–experimentalmethod for the determination of the parameters inthese particular yield criteria was a first step in theapplication of this method for the mechanical charac-terisation of metals at high strains and strain rates. Thestrains and strain rates in the current experiments werestill low, but the sample geometry was non-standard,causing inhomogeneous stress- and strain-fields. In fu-ture research, the behaviour of the material at higherstrain levels will be addressed. Parameters will be esti-mated from measurement data obtained from sheartests or comparable tests [20]. These tests are moresuitable for obtaining higher strains before crack initia-tion than, for example, from tensile tests. Displacementfields are measured on the surface of the samples usingcorrelation techniques [20,21].

Acknowledgements

This work is part of a research project supportedfinancially by the Dutch Technology Foundation(STW).

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