an inverse analysis procedure for the material parameter identification of elastic-plastic...

7/28/2019 An inverse analysis procedure for the material parameter identification of elastic-plastic free-standing foils

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Struct Multidisc OptimDOI 10.1007/s00158-008-0294-8

INDUSTRIAL APPLICATION

An inverse analysis procedure for the material parameteridentification of elastic–plastic free-standing foils

Matteo Ageno · Gabriella Bolzon · Giulio Maier

Received: 10 July 2007 / Revised: 5 March 2008 / Accepted: 11 June 2008© Springer-Verlag 2008

Abstract A model calibration technique is considered

for the estimation of material parameters in free-standing thin foils. The experimental apparatus is in-spired by bursting strength testers for paper, textilefabrics and polymer coatings such as geo-membranes.The procedure referred to herein consists of the fol-lowing phases. A controlled fluid pressure is appliedto the foil specimen placed on an horizontal planewith a suitably shaped hole. The induced out-of-planedisplacements are measured by a laser profilometer.The material parameters are then inferred from thesemeasurements through inverse analysis, by simula-tion of the test and minimisation of a suitable norm

which defines the discrepancy between measured andcomputed displacements. Potentialities and limitationsof the proposed method are assessed on the basisof computer-generated “pseudo-experimental” data,where modelling errors are ruled out. The identifiabilityof some industrially meaningful material parameters isestablished.

Keywords Material characterization ·

Inverse analysis · Free-standing foils ·

Experimental mechanics

M. AgenoThe Italian Pulp and Paper Research Institute (SSCCP),Piazza Leonardo da Vinci 16, 20133 Milano, Italy

G. Bolzon · G. Maier (B)Department of Structural Engineering, Politecnico diMilano, Piazza Leonardo da Vinci 32, 20133 Milano, Italye-mail: [email protected]: www.stru.polimi.it

1 Introduction

Recently, thin foils have been the subject of severalinvestigations especially concerning constitutive mod-els and failure criteria for paper (see, e.g.: Castro andOstoja-Starzewski 2003; Mäkelä and Östlund 2003; Xiaet al. 2003), metallic foils (Banabic et al. 2003; Lemaitreand Chaboche 1994; Hu 2003; Klein et al. 2001), poly-mer films (Bhushan and Ravichander 2004; Lemaitreand Chaboche 1994; Christodolulou 2000; Gerlach andDunne 1994) and laminates (Lange et al. 2002; Wyseret al. 2001). Thin layers are usually anisotropic andcharacterised by an increasing number of materialparameters related to meaningful mechanical prop-erties, which are often influenced by the productionprocess.

A technique for the calibration of these parametersin free-standing thin foils is investigated in what fol-lows, based on the use of experimental instrumenta-tion specifically designed for membrane specimens andintegrated by inverse analysis.

The experimental equipment referred to herein(henceforth called ‘membranometer’) was inspired bybursting strength testers for paper (see e.g. standardsASTM D774/D774M-1997 (Standard test method forbursting strength of paper) and UNI EN ISO 27582004) and for textile fabrics and polymer coatings suchas geo-membranes (see standards ASTM D3786-2001(Standard test method for hydraulic bursting strengthof textile fabrics—diaphragm bursting strength testermethod) and RILEM guidelines, Rollin and Rigo eds.,1991). Membranometric techniques or “bulge tests”have been investigated by Hill and Storakers (1980),Forestier et al. (2003), Ageno et al. (2004) andChamekh et al. (2006).



M. Ageno et al.

The procedure to be considered herein can be out-lined as follows. A controlled fluid pressure is appliedto the foil specimen placed on the horizontal planewith a circular (or otherwise suitably shaped) hole.The induced out-of-plane deformation shape is mea-sured by a laser profilometer. The material parametersare then inferred from these measurements through

inverse analysis, namely: the test is simulated by a fi-nite element technique and out-of-plane displacementsare computed; a norm of the discrepancy betweenmeasured and computed displacements is suitably de-fined and minimised, with respect to the parameters toidentify. Such deterministic, batch (i.e. not sequential)parametric identification is the simplest one among thetraditional methods for material model calibration; see,e.g., the recent surveys by Maier et al. (2006a, b) andStavroulakis et al. (2003).

Recently, Chamekh et al. (2006) trained a neuralnetwork apt to calibrate an elastic–plastic anisotropic

model for metal sheets, using as input data the mea-surement of the maximum bulge displacement forincreasing applied pressure.

The present paper is primarily intended to verifywhether the geometry of a circular bulge can providedata sufficient for the identification of inelastic materialparameters among the ones of industrial interest. Thesensitivities of the measurable quantities with respect tothe sought parameters are computed and the geometryof the, possibly non convex, discrepancy function tominimize is examined in typical situations together withthe robustness of the minimization procedure. It is wellknown, in fact, that inverse problems may suffer of ill-posedness and multiple parameter sets may exist whichpermit to recover the available experimental informa-tion to the same extent (Bui 1994).

In what follows, the above outlined procedure of material characterisation is preliminarily investigatedto these purposes, with reference to inelastic time-independent material models available in a widelyused commercial code. The classical Hill’s criterion foranisotropic materials is considered in the numericalsimulation; see e.g. Lubliner (1990). It reduces to theHencky–Huber–Mises criterion in the special, practi-cally meaningful case of isotropy. Chaboche’s model(see Lemaitre and Chaboche 1994) has been also con-sidered as a possible more realistic alternative for thinisotropic metal foils.

The identifiability of the main parameters in theconsidered material model is herein computationallyinvestigated through the consideration of ‘pseudo-experimental’ information, computer-generated by theforward operator using a priori selected parametervalues; these data are input into the inverse analy-

sis procedure, which thus can be assessed by com-paring its results to the previously chosen values rul-ing out the influence of modeling errors. However,pseudo-experimental data are corrupted by randomlygenerated noise, to take into account the effect of measurement inaccuracy.

Various prospects of the experimental and computa-

tional study now in progress within a broader researchproject in collaboration with the paper industry, areoutlined in the closing section.

2 Experiment simulation

A schematic representation of the experimental equip-ment herein referred to (and called “membranometer”)is shown in Fig. 1. It consists of a container filled with afluid under controlled pressure. A liquid under volumecontrol can be considered as an alternative in order tostabilize the specimen behaviour after possible peaksof pressure. A hole of suitable geometry (circular, inthe present investigations) is open on the upper side of the container. A specimen of the foil to be mechanicallycharacterised is placed on the top of the container andfixed over its opening; this membrane can thereforebe easily inflated by a sequence of increasing pressurelevels. At each level, the out-of-plane displacementsare measured by a laser profilometer and recorded.These quantities are input data for the parameter iden-tification procedure based on computer simulation of

the experiment and on minimisation of the discrepancyfunction. As a main improvement with respect to theconventional bulge test, the geometry of the whole in-flated membrane can be monitored to enrich the set of

Fig. 1 Schematic representation of the instrumentation calledmembranometer



An inverse analysis procedure for the material parameter identification of elastic–plastic free-standing foils

experimental information available to inverse analysispurposes.

In what follows large strains are considered for thelocal mechanical behaviour of the inflated thin paperor metal foils, and large displacements are allowedfor in view of the significant influence of geometrychanges on equilibrium. A finite element model apt

to simulate the above experimental test is generatedthrough a commercially available finite element code(here Abaqus/Standard 2006) in its version conceivedfor geometric nonlinearities, which, therefore, will notbe discussed herein.

Clearly, as long as the hole radius is by ordersof magnitude larger than the foil thickness, biax-ial stress states are dominant in the specimen; sincebending effects are confined to a narrow strip alongthe circumferential boundary, a membrane state of stress (σ 33 = σ 13 = σ 23 = 0) is assumed everywhere. An

analytical solution of the circular isotropic membraneanalysis problem was achieved by Hill and Storakers(1980) in small-deformation anelastic regimes.

In-plane orthotropic behaviour is attributed here tothe foil material. In the elastic range, Hookean stress–strain relationship, containing four parameters ( E1, E2,G12, ν12), is adopted:⎧⎨⎩ε11

ε22

ε12

⎫⎬⎭=

⎡⎣ 1

E1 − ν12

E2 0

−ν12

E2 1

E2 0

0 0 1

2G12

⎤⎦⎧⎨⎩σ 11

σ 22

σ 12

⎫⎬⎭ (1)

In isotropic elasticity the independent parameters re-duce to two, namely: E1 = E2 = E; ν12 = ν21 = ν;G12 = E

2 (1 + ν).

The elastic domain of an elastic–plastic orthotropicmaterial can be determined by the classical Hill’s cri-terion (see e.g. Lubliner 1990). In plane-stress space, itcan be described analytically by the inequality:

f =

σ 11

R11

2

+

σ 22

R22

2

+ 3

σ 12

R12

2

− σ 11σ 22

1

R2

11

+1

R2

22

−1

R2

33

− σ 0 ≤ 0 (2)

where the material parameters Rii > 0 (i = 1, 2, 3) rep-resent the ratios, with respect to a conventional valueσ 0, of the yield limits (equal in tension and in compres-sion) for uniaxial stress along the orthotropy axes (sameas for the elastic behaviour); R12 > 0 is the analogousratio for in-plane shear yield stress.

Under isotropy hypothesis, i.e. with Rii = 1 andR12 = 1, the yield condition (2) reduces to Hencky–Huber–Mises (HHM) criterion. In this case, σ 0 canbe interpreted as the yield limit in uniaxial tension/compression tests.

To ensure the existence of Hill’s yield function (2)for any membrane state of stress, coefficients Rii mustdefine a positive-definite quadratic form; therefore, ra-tios r 22 = R22/R11 and r 33 = R33/R11 must belong tothe dashed region shown in Fig. 2, which satisfies theinequalities:

r 22r 33 + r 22 − r 33 > 0; r 22r 33 − r 22 − r 33 < 0;

r 22 r 33 − r 22 + r 33 > 0 (3)

In some technological fields, particularly in metal form-ing, Lankford coefficients are employed to describeanisotropy as the ratio between transversal strains in aspecimen subjected to uniaxial tension beyond the elas-tic limit. These parameters can be correlated to Hill’sratios Rij as shown, e.g., by Khalfallah et al. (2002)

or in the users’ manuals of the finite element com-mercial code here employed (Abaqus/Standard 2006).Lankford coefficients may be evaluated according toASTM E 517-00 (Standard test method for plastic strainratio r for sheet metal), for metal sheets, but the workby Chamekh et al. (2006) has demonstrated that theircalibration based on the data of tensile tests does not

return satisfactory predictions of the material responseto multi-axial loading.

Fig. 2 Theexistence domain of Hill’s yield function ( gray region)



M. Ageno et al.

Stress and strain rates are assumed herein to be gov-erned by the usual relationships of associative elasto-plasticity:

ε̇ij = ε̇eij + ε̇

pij (4)

ε̇eij = C ijrsσ̇ rs, ε̇

pij = λ̇

∂ f

∂σ ij (5)

f ≤ 0 , λ̇ ≥ 0 , f λ̇ = 0 (6)

In the above formulae: εij represent the strain com-ponents (those listed in (1) plus the component ε33

associated with the membrane thickness variation),additionally decomposed into their elastic (reversible)εe

ij and inelastic (plastic) εpij addends; σ rs and C ijrs are

components of the stress tensor and entries of the(constant) elastic compliance tensor in (1), respectively;λ denotes the plastic multiplier; dots mark rates, namelyderivatives with respect to sequential (not physical)time; multiplication of quantities with repeated in-dices implies summation with respect to them. Thecomplementarity relations (6) reflect the irreversibility(history-dependence, nonholonomy) typical of plasticbehaviour.

In the numerical tests presented in what follows,perfect plasticity is assumed beyond the elastic limit foranisotropic metal foils.

In isotropic elastoplasticity, exponential hardeningwill be considered here beyond the original elastic

range defined by (2). The adopted hardening rule isdescribed by the following relationship that governsthe yield stress σ ≥ σ 0 in uniaxial stress states and isendowed with two plastic parameters (σ 0 and σ ∞):

σ = σ ∞ − (σ ∞ − σ 0) e−

Eε−σ 0σ ∞ −σ 0

(7)

The initial slope of the stress σ versus total strain curveis forced by (7) to be equal to the elastic modulus E, sothat σ (ε) is “smooth” curve. In terms of plastic strainεp, (7) becomes:

σ = σ ∞ − (σ ∞ − σ 0) e−

σ −σ 0 +Eεp

σ ∞−σ 0 (8)

This equation is an implicit definition of the yield stressσ which, for given εp, can be evaluated by a solving rou-tine. In multiaxial stress states εp acquires the meaningof Mises equivalent plastic strain and (8) replaces theinitial yield limit σ 0 in (2).

As an alternative to the above material model, theassociative elastic–plastic–hardening isotropic constitu-

tive law will also be considered. Its formulation (seeLemaitre and Chaboche 1994), specialised to kinematichardening only, can be expressed by the preceding raterelationships (4) to (6), and by the following relationswhich replace (2) and (7 and 8):

f =

3

2σ

ij − X ij

σ

ij − X ij

− σ

2

0 ≤0

(9)

˙ X ij =2

3C ̇ε

pij − γ X ij

2

3ε̇

prs ε̇

prs (10)

In the above formulae: σ ij is the deviatoric part of the stress tensor σ ij ; f ≤ 0 defines the current elasticdomain, centred on point X ij in the stress space (the so-called “back-stress” tensor); C and γ are the materialparameters that control hardening evolution accordingto (10).

In monotonic uniaxial-stress situations, closed-formintegration of the above relationships provides the fol-lowing exponential hardening function of the plasticdeformations, as an alternative to (7):

σ = σ 0 +C

γ

1 − e−γ εp

(11)

where εp denotes the equivalent plastic strain (whichreduces to the axial plastic strain in uniaxial stressstates). Equation (11) exhibits the dependence of Chaboche kinematic hardening curve by three indepen-

dent parameters: initial yield stress σ 0

, initial slope of the hardening curve C , and saturation value of the yieldstress C /γ .

3 Parameter identification

A simple deterministic non-sequential (batch) least-square approach is adopted herein for the identificationof material parameters, namely the available experi-mental information are exploited all together at theend of each test. Uncertainties affect both the measure-

ments and the system modelling. In what follows, theeffects of noisy input data on the estimates will be in-vestigated only to the purpose of evaluating the robust-ness of the parameter calibration procedure; howevermodelling errors will be systematically ruled out for thepresent preliminary validation purposes.

In the experiment the membrane is inflated by asequence of M pressure steps, run by index, h = 1,




2. . .M typically M = 5 ÷ 10. Elastic parameters areidentified first, at low pressure, and plastic parameterslater, at higher pressure (Section 5). At each load-ing step, the out-of-plane (vertical) displacements de-noted by um

hk (k = 1, 2 . . . N ) of N selected points on themembrane surface, are supposed to be measured andrecorded by means of an ad-hoc instrument such as a

laser profilometer.The discrepancy between the measured displace-

ments and the corresponding computed quantitiesmarked by superscript c, uc

hk with, k = 1, 2. . .N isquantified by the following “discrepancy function”,namely by the Euclidean norm of the “discrepancyvectors”, a norm which represents the objective func-tion to minimise under the constraints defined, e.g., byrelations (3):

ω (x) =

M

h=1

N

k=1

uchk (x) − um

hk

um

h max

2

(12)

where vector x collects the unknown parameters tobe identified through the discrepancy minimisationprocess. All displacements at each pressure level h arenormalised by the maximum displacement (measuredin the bulge centre) at that pressure step, um

h max. The

dependence of the computed quantities on the parame-ter vector x is implicitly defined by the constitutive re-lationships adopted in the FE model. This dependencemakes the objective function ω non-explicit and gener-ally non-convex function of x. Usually, the discrepancy

function is generated by the quadratic form associatedto the inverse of the covariance matrix of measure-ments. In the present context this matrix is assumedas identity, since the experimental data provided bythe same instrument can be regarded as uncorrelatedand affected by the same standard deviation of randomerrors.

The constrained minimisation of the above objectivefunction ω(x) in (12) can be performed by a number of numerical methods. Among those available in the pop-ular Matlab (2002) optimisation toolbox, the SequentialQuadratic Programming (SQP) and Trust Region (TR)

algorithms have been selected for the present purposesafter some comparative numerical tests.

The SQP algorithm coded in Matlab package is de-signed to numerically solve general constrained optimi-sation problems, with recourse to the Lagrangian whichincorporates the objective function and the constraints,including those formulated in (3). A sequence of con-vex quadratic approximations to this Lagrangian is built

up by using information from gradients only (no secondderivatives are evaluated). At each iteration the globaloptimum point is computed (by linear search) recur-sively along the direction which minimises the convexquadratic approximations constructed in the currentiteration. This algorithm has been efficiently employedin a number of different constrained optimisation prob-

lems; see, e.g., Bolzon et al. (1997), Maier et al. (2001),Stavroulakis et al. (2003).

The alternative iterative first-order TR algorithmcan be efficiently employed for large scale problemswhen box-constraints only are defined on the minimi-sation variables, as for the case of isotropic materialsin the present context. Each iteration step is centredon a two-dimensional quadratic programming problemin the plane defined by the gradient of the objectivefunction and by its conjugate Gauss–Newton direction.

A detailed description of these algorithms, omittedhere for brevity, can be found in Matlab manuals and

in publications referenced therein.In practical applications, the solution procedure

starts from suitably chosen initial estimates of thesought parameters, either previously assessed on bulkmaterial or expected on the basis of handbooks, pre-vious experience or expert’s judgement. The “exact”values of the sought parameters are a priori assumed inSection 5, for the validation of the proposed methodol-ogy and for the preliminary assessment of its potential-ities and limitations. In the numerical tests summarizedin Section 5, each parameter will be initialised at thehalf of its “exact” value, i.e. far away from it, in orderto test the robustness of the algorithms.

4 Sensitivity analysis

The design of the experiments to be combined withparameter identification procedures may be orientedand enhanced by preliminary sensitivity analyses. Suchanalyses are intended to quantify the influence of eachsought parameter on measurable quantities and, hence,to corroborate the expectation on its identifiability (see

e.g., Kleiber et al. 1997; Stavroulakis et al. 2003).In the present context, the sensitivities of the out-

of-plane displacements with respect to the materialparameters at a given pressure level, have been repeat-edly computed by a central finite difference scheme.To simplify comparisons, the approximated deriva-tives have been normalised with respect to the consid-ered displacement uhk (h = 1, 2. . .M ; k = 1, 2. . .N ) and



M. Ageno et al.

material parameter, so that the sensitivity indices of the measurable displacement, with respect to the elastic

moduli E and ν, read (similarly for other parameters toidentify):

sE =E

uchk

∂uchk (x)

∂E

x=x

∼=E

2E

uchk

E + E, ν ,σ 0, . . .

− uc

hk

E − E, ν ,σ 0, . . .

uc

hk E, ν ,σ 0, . . . sν =

ν

uchk

∂uchk (x)

∂ν

x=x

∼=ν

2ν

uchk

E, ν + ν,σ 0, . . .

− uc

hk

E, ν − ν,σ 0, . . .

uc

hk

E, ν ,σ 0, . . .

(13)

Clearly, in real-life situations, vector x contains para-meter estimates preferably representative of the inter-val considered, or a priori conjectured by an “expert”.

In the present computational validation of the pro-posed inverse analysis procedure (Section 5), the

argument x = E, ν ,σ 0, . . . of the derivative ap-

proximated according to (13), is assumed equal tothe “exact” value originally adopted for computingthe pseudo-experimental data. The increments (E,

ν,. . .) have been set equal to 10−3 ·

E, ν , . . .

after

a preliminary investigation on the influence of numer-ical round-off errors on the performance of the finitedifference approximations of derivatives.

Some selected results of the performed sensitivitystudies are presented and commented on in the nextSection.

5 Numerical results

The membranometric technique outlined in what pre-cedes for the identification of material parameters infree-standing foils will be numerically tested in thissection with reference to both isotropic and anisotropicfoils. As pointed out in the Section 1, reasonablychosen values are attributed to the parameters andlead, through the “forward operator”, to “pseudo-experimental” information, i.e. to values of the mea-surable quantities generated by the FE model. Thesepreliminarily computed data replace truly experimentalinformation in the present applications, so that com-parisons of expected (a priori known) and actuallyobtained values permit to check the identifiability of the model parameters based on the proposed instru-mentation and inverse analysis technique. The pseudo-experimental data are fed into the inverse analysisprocedure after being corrupted by random “noise”,generated with uniform probability density over agiven interval, in order to check the robustness of theproposed identification technique with respect to

measuring and modelling errors. In the following com-putational exercises, the TR algorithm has been exten-sively used to optimisation purposes.

Pseudo-experimental data are generated by meansof finite element modelling, considering a circular win-dow of 10 cm diameter. The space discretization rests

on the mesh shown in Fig. 3 and on 300 bilinear(four nodes) elements (892 degrees of freedom) imple-mented in Abaqus/Standard (2006). The model reducesto 20 elements defining an inflated profile in the case of isotropic materials and consequent axial symmetry. Inthe present validation exercises and in most expectedindustrial situations, the ratio between the window di-ameter and the specimen thickness reaches 2 or 3 or-ders of magnitude: this circumstance makes obviouslylegitimate to neglect the flectural effects confined to

Fig. 3 Finite element mesh employed for the simulation of theinflating process of a membrane placed over a circular membra-nometer window




Table 1 Apex displacement (in mm) resulting from differentspace discretizations

Elastic model Elastic–plastic model

Membrane, 20 elements 4.037345 14.652947Membrane, 50 elements 4.036544 14.656882Membrane, 200 elements 4.031884 14.659005Shell, 20 elements 4.037352 14.653697Shell, 50 elements 4.036543 14.659886

Shell, 200 elements 4.036381 14.660255

a narrow layer near the boundary and to everywhereassume membrane state (plane-stress, constant alongthe foil thickness). Whenever required by specific ap-plications (for diameter/thickness ratio unusually low),a shell model may replace the membrane one, as donee.g. by Chamekh et al. (2006), without substantialchanges in the methodology employed herein.

The adopted mesh has been tested in order to assesspossible model and mesh sensitivity of the solution.

The computational exercises have concerned: elasticisotropic material with parameters E = 35, 000 MPa,ν = 0.33, under inflating pressure equal to 10 kPa;isotropic elastoplastic Mises material with the sameelastic parameters and yield limit σ 0 = 35 MPa underpressure equal to 6 kPa. In both cases, foil thickness wasassumed equal to 9 μ m. Simulations performed by usingdifferent refinements and element types (with mem-brane elements pinned to the boundary, shell elementsclamped to it) led to the apex displacements gatheredin Table 1.

These results show that the mesh-sensitivity is con-

fined to the fourth (and subsequent) digit. Therefore,to the purposes pursued in this paper, a coarse meshis acceptable, and useful to reduce the computationalburden. It is worth noting that the elastic models turnout to become stiffer by increasing the number of degrees of freedom; this effect covers the relaxationdue to the increase of the number of degrees of free-dom; this circumstance is due to the solution regular-ization algorithm embedded in the employed FE code(Abaqus/Standard 2006).

5.1 Isotropic elastic–plastic foils

A 9 μ m thick aluminium foil is considered first as areference industrial case. For the numerical validationtests of the present technique, isotropic material behav-iour is initially assumed. In view of the consequent ax-isymmetry of the bulge configuration the displacementsof the 20 free nodes along a radius are dealt with asinput for inverse analysis.

Due to the kinematic boundary conditions (displace-ments and, hence, circumferential deformation con-

Fig. 4 Stress distribution along the radius, of an isotropic foilin the elastic range (low pressure): σ 11 = radial stress, σ 22 =

circumferential stress

strained along the border) and to the initially planarconfiguration, the inflated membrane assumes a non-spherical shape and a non-uniform stress distribution,as shown in Fig. 4 concerning a situation in the elasticrange. In other terms, the static regime in the bulgingmembrane turns out to be substantially different fromthe uniform statically determined one in a sphericalinflated bubble and classical Mariotte’s spherical seg-ment, as shown by Hill and Storakers (1980) in smalldeformation regimes. This circumstance makes it pos-sible to identify both E and ν separately, in tests withabsence of plastic deformations.

Beyond the elastic range, the Hencky–Huber–Mises(HHM) model of “perfect” plasticity (no hardening)is considered first as a particularisation to isotropy of constitutive relations (1)–(6). The assumed materialparameters are the following ones, typical of laminatedaluminium: Young modulus E = 45 GPa; Poisson ratio

Fig. 5 Vertical displacements, at different pressure levels, alongthe radius in a foil of an isotropic material which behaves accord-ing to HHM plasticity model without hardening



M. Ageno et al.

Fig. 6 Displacementsensitivity indices, normalizedaccording to (12), andconcerning Chaboche modelparameters (E, ν, σ 0, C andγ ) at the pressure levels p

specified in the table

ν = 0.3; yield limit σ 0 = 35 MPa. The vertical displace-ments at various pressure levels, computed along a ra-dius at the nodes of the FE mesh (Fig. 3) with the aboveperfectly-plastic material model are shown in Fig. 5.The equivalent plastic strain exhibits its maximum atthe apex node and grows with the pressure up to a

critical ductility limit (here assumed 10%), under themaximum imposed pressure (here 5.1 kPa).

After the above classical material model with a sin-gle plasticity parameters, the numerical validation of the proposed technique is performed with reference topopular, still isotropic, models endowed with two and

Fig. 7 Convergence patternof the two-phaseidentification procedure for

isotropic perfectly plasticHHM model: a elastic andb inelastic phase. In c, with asingle inverse analysis, lackof correct convergence isexhibited for Poisson ratio




three plasticity parameters and concerning two otherkinds of metallic foils, namely:

(i) HHM model with exponential isotropic hardeningdescribed by (7) and defined by the yield limitσ 0 = 35 MPa and ultimate strength σ ∞ = 70 MPa;

(ii) Chaboche model, (9)–(10), with kinematic hard-

ening, see (11), defined by the three parame-ters σ 0 = 35 MPa, C = 20 GPa and γ = 1,000.The monotonic (non-cyclic) behaviour of the testmakes in fact immaterial (and not identifiable)the difference between kinematic and isotropichardening envisaged by Chaboche model.

Specimens are regarded as being cut from the con-sidered metal foils and subjected to a sequence of pressure levels. In this first validation exercise, thenumbers of pressure levels considered for collect-ing pseudo-experimental displacements are M = 5 forHHM models (with and without hardening) and M = 6

for Chaboche model.The sensitivity indices of measurable vertical dis-

placements defined in the preceding Section 4, withrespect to the five parameters of Chaboche model, areplotted in Fig. 6, evaluated at the pressure levels speci-fied in the same figures. It can be noticed that the sen-sitivity is almost constant along the membrane radiusfor all parameters except Poisson ratio (which exhibitsthe least influence on measurable displacements) andthat it obviously drops down at the boundary wheredisplacements are constrained. Sensitivities of the mea-surable displacements are quite large and comparableat the lowest pressure ( p = 0.6 kPa) with respect to theelastic parameters (Young modulus E and Poisson ratioν); they become almost negligible for p = 2.1 kPa. Onthe contrary, as expected, sensitivity grows at higherpressure levels ( p ≥ 2.91 kPa) with respect to yieldstress and hardening parameters C and γ .

Numerical exercises corroborated by the preced-ing computations of sensitivities, suggest the follow-ing identification strategy: elastic moduli are estimatedfirst, on the basis of membrane responses to low pres-sure levels; these values are then input into the finiteelement model for a second identification stage, whichdeals with the inelastic material parameters. Figure 7a,b show the rather fast convergence toward the “exact”values, when the above two-step procedure is carriedout with the assumption of “perfect” plasticity. Eachparameter is normalised by its originally assumed value,so that convergence toward the common value 1 isexpected from successful inverse analyses. Only datarelevant to the lowest pressure level are employed inthe first (elastic) identification stage; the displacementsunder the further levels are used to determine the

Fig. 8 Convergence patterns of the plastic parameters: a in HHMmodel with exponential isotropic hardening (parameters σ 0 andσ ∞); b in Chaboche model with kinematic hardening (parametersσ 0, c and γ )

yield limit. Figure 7c visualises the unsatisfactory resultsobtained by considering the pseudo-experimental dataand the unknown parameters all together: the identifi-cation of Poisson’s ratio fails, as expected in view of lesssensitivity of measured displacements with respect to it.

Figure 8a, b show the convergence pattern of theplastic parameters, in the second phase of the two-stepidentification procedure, for the two alternative optionsearlier mentioned for hardening elasto-plasticity. Re-sults are in both cases quite satisfactory in terms of con-verged estimates, but much larger iteration numbers(and, hence, larger computational burdens) are worthnoting for the three plastic parameters of Chabochemodel than for the two ones of HHM model.

The robustness of the proposed procedure has beenfurther tested by means of “noisy” input data. Specif-ically, the pseudo-experimental displacements havebeen perturbed before being considered input datafor the identification process. The noise consists of a random addendum generated with uniform proba-bility density over various intervals ±, up to 5%



M. Ageno et al.

Fig. 9 Converged values of Chaboche model parametersestimated ( gray dots) andtheir averages (white dots),under the pressure levelsspecified in Fig. 5, on thebasis of data corrupted bygrowing amplitude of randomperturbation: a elastic

modulus E; b yield stress σ 0;c Poisson’s ratio; d initialhardening slope C ;e kinematic hardeningparameter γ . Dashedlines represent the 95%confidence boundary

discrepancy with respect to the “exact” displacementamplitude.

Figure 9a to e show the estimates (after con-vergence) of scatter bands of the five parametersin Chaboche model, which have been evaluated forthe pressure levels indicated in Fig. 5, on the basisof artificially corrupted noisy data, namely: throughten identifications initialized by ten different randomrealisations of the same noise level. For most para-meters, estimation errors turn out to remain withinthe relative ranges of the input noise. Accuracy of the estimates can be increased by exploiting furtherexperimental data (of course with increasing compu-tational efforts). This circumstance is quantified inFig. 10, where the scatter bands (once again based onten random realisations) of converged estimates of theyield stress σ 0 in HHM perfect plasticity are plotted forvarious input noise (specified there) as functions of the

increasing number of pressure levels and, therefore, of displacement measurements employed in the inverseanalysis. It can be seen how the growth of the avail-able experimental information progressively reducesthe uncertainty of the identified material parameters. Inparticular, it turns out that the estimation uncertaintyremains within the ranges of the input relative errorswhen data from at least two loading steps (pressurelevels) are used to identification purposes.

5.2 Anisotropic elastic–plastic foils

An orthotropic material model will now be considered,characterised by the following set of in-plane elastic pa-rameters according to (1): E1 = 40 GPa; E2 = 25 GPa;G12 = 18 GPa; ν12 = 0.3. These values are typical of thin aluminium foils used in food package industry.Anisotropy due to production processes, specifically




Fig. 10 Converged normalised estimates of the elastic limit (pa-rameter σ 0) in HHM perfect plasticity, versus the number of pressure levels employed in the identification procedure. Noisyinput data are generated with uniform probability density in theintervals ±, with = 0.5%,1%,2%,5%. Dashed lines representthe scatter band of 95% confidence boundary

orthotropy in “machine direction” (in some technolog-ical fields also called “rolling direction”) and “crossdirection”, is a frequent and sometimes technologicallyimportant feature of laminated sheets and foils.

Pseudo-experimental data have been generated as-suming perfectly plastic behaviour (no hardening) andHill’s orthotropic yield criterion (2), quantitativelycharacterised by the limit stressσ 0 = 35 MPa and bytheratios: R11 = 1; R22 = 0.8; R33 = 0.75; R12 = 0.7.

Hill’s equivalent stress distributions along the twoprincipal radii are plotted in Fig. 11 for the variouspressure levels specified there. It can be observed thatthe onset of plastic yielding occurs at the apex of theinflated membrane. The same was noticed in numericalsimulation of isotropic membrane inflation. However,numerical exercises with different parameter combina-tions evidenced the contrary circumstance, i.e. plasticdeformations beginning at the boundary.

The material anisotropy, clearly, is reflected by dis-placement radial distributions, which vary with the ra-dius orientation, as visualised in Fig. 12 for one pressurelevel ( p = 4.8 kPa). Many more displacements have

Fig. 12 Difference between displacements along differently ori-ented radii (90◦ correspond to the x2 axis) and the same along thematerial direction with maximum Young modulus (i.e., x1 axis inFig. 2), under pressure p = 4.8 kPa

now to be used as input data in the inverse analysisprocedure. Specifically, the following parameter identi-fication exercises will be based on the N = 279 verticaldisplacements at all free nodes in the FE mesh shownin Fig. 2.

Like in the previous isotropic case, elastic moduliare identified first by a single pressure level; thereafter,the four parameters governing plastic behaviour areestimated all together by inverse analysis based ondisplacements under subsequent four pressure levelsapplied in a sequence.

The convergence patterns of the elastic parametersare shown in Fig. 13a, b respectively, for the cases of “exact” pseudo-experimental and noisy displacements.Fast convergence toward almost the exact solution oc-curred in the absence of noise (Fig. 13a): a maximum1.2% residual error is observed on the converged G12

value. In the parameter estimation exercise based onnoise-corrupted displacements (affected by a rathersevere random disturbance with constant probabilitydensity over the interval ±5%), it turns out that onlythe consequent error on the Poisson ratio ν12 (∼7%)exceeds the level of the input disturbance.

Fig. 11 Equivalent Hill

stress, normalised by thereference yield stress, atvarious pressure levels p,along radial axes: a in thestrongest direction (withR11 = 1); b in the weakestdirection (R22 = 0.8)



M. Ageno et al.

Fig. 13 Identification of theelastic parameters of anorthotropic membrane:a by noiseless “exact”pseudo-experimental data;b with random perturbationsin the interval ±5%

Fig. 14 Estimation of plastic parameters in Hill perfectly-plastic model of an orthotropic membrane: a ratios R11, R22 and R33 withfixed R12; b ratios R11 and R22 with both R12 and R33 a priori known; c ratios R12 and R33 with fixed R11 and R22 (details in the insert )

Fig. 15 Estimation of parameters in Hill anisotropicmodel: a contour plot of thelogarithm of the discrepancyfunction, with R11 and R22

fixed to their “exact” values;b logarithmic plot of thediscrepancy function aroundits absolute minimum point,with R33 fixed to its exactvalue: thick line concerns thesame computation grid like ina; dotted line refers to a morerefined grid




Finally, the performance of the mechanical charac-terisation procedure has been checked by carrying outthe identification of the four plasticity parameters in(2). The uniaxial yield limit in the stronger direction,is assumed as reference value σ 0. Figure 14a shows theconvergence patterns when the shear parameter R12

is fixed at its “exact” value: the identification of R11,

R22 and R33 fails. If also R33 is a priori known, thenit is possible to recover the exact R11 and R22 valuesas evidenced by the results plotted in Fig. 14b. Anidentification process of R12 and R33 with R11 and R22

fixed at their exact values is visualised in Fig. 14c: about5% residual error for the shear limit R12 is obtainedeven in this case of “exact” data. It can be conjecturedthat the lack of convexity of the discrepancy functionand the existence of local minimum points do not allowthe exact determination of all the sought parameterseven in the absence of experimental noise.

This interpretation of the preceding results is cor-

roborated by the plots in Fig. 15, where the existenceof local minima in the vicinity of the global minimumis evidenced. These plots have been obtained by 533repeated direct analyses varying the parameter valuesin the grid defined by 41 × 13 points, in the R12 andin the R33 axis direction, respectively. In the presentnumerical test it turns out that the absolute minimumcorresponding to the “exact” parameter set is orders-of-magnitude lower than any other local minimum of the discrepancy function. Global searches based, e.g.,on evolutionary algorithms (Tin-Loi and Que 2002)might capture this absolute minimum. However, inthe presence of modelling and/or experimental noise,the distinction among the different minimum values of the discrepancy function can be much mitigated.

Fig. 16 Displacement sensitivity indices, normalized accordingto (12), concerning Hill’s model parameters in plane stress prob-lems (R11, R12, R22 and R33) at the highest pressure leveladopted in the present identification exercise

It can be concluded that, to engineering purposes,the calibration procedure is effective for the two plasticparameters which are most meaningful, namely theyield limits (represented by the ratios R11 and R22)

in the membrane in-plane principal directions. Theirdominancy is confirmed by the results of sensitivityanalyses shown in Fig. 16.

6 Closing remarks and future prospects

A procedure centred on a novel instrument calledmembranometer, and inspired by bursting strengthtests on paper and/or coated fabrics such as geo-membranes, has been investigated in view of its rou-tine use in industrial environments for the mechanicalcharacterisation of free-standing thin films and foils ina simple and economical manner.

The procedure basically consists of the measure-ment, by a laser profilometer, of the shape of an inflatedmembrane specimen located over a circular hole, to beinserted into an inverse analysis for the identificationof parameters contained in the material model selectedfor engineering applications. Potentialities and limi-tations of such methodology have been preliminarilyassessed.

The computational tests performed to validationpurposes have concerned time-independent inelasticmaterial models and have employed a simple deter-ministic batch technique for inequality constrainedminimization.

Satisfactory performance was observed in the caseof isotropic materials: yield limits and hardeningcoefficient can be identified in a robust manner.

The elastic characterisation of anisotropic materialmodels was verified in a quite straightforward way.In the plastic range, the calibration of the classicalanisotropic Hill’s criterion was shown to be quite possi-ble and effective for the two parameters which are mostmeaningful, namely the yield limits in the membrane in-plane principal directions. Estimations of the remainingtwo plastic parameters turned out to be difficult orimpossible due to multiplicity of local minima of the(non convex) discrepancy function.

The following issues are worth being pursued by theresearch in progress on the membranometer method-ology: (a) comparative study of various die aperturegeometries, alternative to the circular one; (b) direct-search genetic algorithms as mathematical tools alter-native (at least in a first stage) to the mathematicalprogramming employed herein and apt to overcomethe above limitations due to the existence of local



M. Ageno et al.

minima; (c) assessment of modelling errors and theirconsequences; (d) processing uncertainties by MonteCarlo or Kalman filtering approaches (see e.g. Bittantiet al. (1983) and Bolzon et al. (2003)); (e) use of suit-ably designed and “trained” artificial neural networksfor routine industrial applications, in order to confinethe expensive large-scale computations to the “train-

ing” and “testing” phases, carried out once for all inthe equipment production, as in Maier et al. (2006c);(f) extension of all the above procedures to fracturemechanics and time-dependent (viscous) behaviours inorder to assess the relevant material properties.

Diverse industrial situations which might benefit bymembranometric tests integrated by inverse analysisare mentioned here in closing: (i) characterisation of the mechanical properties of free-standing (few mi-crons thick) metallic foils to be employed in laminatesfor food and beverage container production; (ii) me-chanical characterisation of paper, paperboard, com-

posite laminates, of which also strength and fractureproperties are of interest; (iii) calibration of parame-ters in visco-elastic and visco-elasto-plastic materialmodels for polymer foils, like geo-membranes in damengineering, by means of deformation measurementsrepeated at suitable time intervals on the membranespecimen under time-constant pressure; (iv) charac-terisation of organic tissues of biomedical interest, if necessary tested in suitably controlled environment.

Acknowledgements A research grant from SSCCP is gratefullyacknowledged.

References

Abaqus/Standard (2006) Theory and user’s manuals, release 6.6.Hibbitt, Karlsson and Sorensen, Pawtucket, RI, USA

Ageno M, Bocciolone M, Bolzon G, Cigada A, Maier G, ZappaE (2004) Mechanical characterisation of thin foils by pro-filometer measurements of inflated membranes. In: Pro-ceedings of 12th international conference on experimentalmechanics, ICEM12, Bari, Italy, August 29–September 2

Banabic D, Kuwabara T, Balan D, Comsa DS, Julean D (2003)Non-quadratic criterion for orthotropic sheet metals underplane stress condition. Int J Mech Sci 45:797–811

Bhushan B, Ravichander J-R (2004) Modeling of creep andshrinkage behavior of polymeric films used in magnetictapes. J Appl Polym Sci 91:78–88

Bittanti S, Maier G, Nappi A (1983) Inverse problems in struc-tural elasto-plasticity: a Kalman filter approach. In: SawczukA, Bianchi G (eds) Plasticity today. Elsevier, London, UK,pp 311–329

Bolzon G, Ghilotti D, Maier G (1997) Parameter identificationof the cohesive crack model. In: Sol H, Oomens CWJ (eds)Material identification using mixed numerical experimentalmethods. Kluwer Academic, Dordrecht, pp 213–222

Bolzon G, Fedele R, Maier G (2003) Parameter identification of a cohesive crack model by Kalman filter. Comput MethodsAppl Mech Eng 191:2847–2871

Bui HD (1994) Inverse problems in the mechanics of materials:an introduction. CRC, London

Castro J, Ostoja-Starzewski MO (2003) Elasto-plasticity of paper.Int J Plast 19:2083–2098

Chamekh A, BelHadj Salah H, Hambli R, Gahbiche A (2006) In-verse identification using the bulge test and artificial neuralnetworks. J Mater Process Technol 177:307–310

Christodolulou K, Hatzikiriakos SG, Vlassopoulos D (2000) Sta-bility analysis of film casting for PET resins using a mul-timode Phan-Thien–Tanner constitutive equation. J PlastFilm Sheeting 16:312–332

Forestier R, Ben Tahar M, Chastel Y, Massoni E (2003) Applica-tion of an inverse method to the analysis of the bulge test.In: Proceedings Esaform conference on material forming.Salerno, Italy, pp 343–346, April 28–30

Gerlach CFG, Dunne FPE (1994) Elastic–viscoplastic large de-formation model and its application to particle filled polymerfilm. Comp Mater Sci 3:146–158

Hill R, Storakers B (1980) Plasticity and creep of pressur-ized membranes: a new look at the small-deflection theory.

J Mech Phys Solids 28(1):27–48Hu W (2003) Characterised behaviours and correspondingyield criterion of anisotropic sheet metals. Mater Sci EngA345:139–144

Khalfallah A, Bel Hadj Salah H, Dogui A (2002) Anisotropicparameter identification using inhomogeneous tensile test.Eur J Mech A Solids 21:927–942

Kleiber M, Antúnez H, Hien TD, Kowalczyk P (1997) Parametersensitivity in nonlinear mechanics. Theory and finite elementcomputations. Wiley, Chichester

Klein M, Hardboletz A, Weiss B, Khatibi G (2001) The ‘sizeeffect’ on the stress–strain, fatigue and fracture propertiesof thin metallic foils. Mater Sci Eng A319:924–928

Lange J, Haroun M, Wyser Y (2002) Understanding puncture re-sistance and perforation behaviour of packaging laminates.

J Plast Film Sheeting 18:231–244Lemaitre J, Chaboche J-L (1994) Mechanics of solid materials.Cambridge University Press, UK

Lubliner J (1990) Plasticity theory. Macmillan, New York, USAMaier G, Bolzon G, Tin-Loi F (2001) Mathematical program-

ming in engineering mechanics: some current problems,Chapter 10. In: Ferris MC, Mangasarian OL, Pang J-S (eds)Complementarity: applications, algorithms and extensions.Applied optimization series, vol. 50. Kluwer Academic,Dordrecht, The Netherlands, pp 201–232

Maier G, Bocciarelli M, Fedele R (2006a) Parameter identifi-cation by inverse analysis in structural mechanics. IACMExpressions 19:21–28

Maier G, Bocciarelli M, Bolzon G, Fedele R (2006b) Inverseanalyses in fracture mechanics. Int J Fract 138:47–73

Maier G, Fedele R, Miller B (2006c) Health assessment of con-crete dams by overall inverse analyses and neural networks.Int J Fract 137:151–172

Mäkelä P, Östlund S (2003) Orthotropic elastic–plastic mater-ial model for paper materials. Int J Solids Struct 40:5599–5620

Matlab (2002) User’s guide and optimization toolbox user’s guide(version 3). Release 6.13. Math Works, USA

Rollin A, Rigo JM (eds) (1991) Geomembranes, identificationand performance testing. RILEM Report 4, Chapman andHall, London, UK




Stavroulakis G, Bolzon G, Waszczyszyn Z, Ziemianski L (2003)Inverse analysis, Chapter 13. In: Milne I, Ritchie RO,Karihaloo B (eds) Comprehensive structural integrity, vol. 3:numerical and computational methods (R. de Borst, H.A.Mang, eds.). Elsevier, Oxford (UK), pp 685–718

Tin-Loi F, Que NS (2002) Identification of cohesive crack frac-ture parameters by evolutionary search. Comput MethodsAppl Mech Eng 191:5741–5760

UNI EN ISO 2758 (2004) Paper—determination of burstingstrength

Wyser Y, Pelletier C, Lange J (2001) Predicting and determiningthe bending stiffness of thin films and laminates. PackagTechnol Sci 14:97–108

Xia QS, Boyce MC, Parks DM (2003) A constitutive modelfor the anisotropic elastic–plastic deformation of paper andpaperboard. Int J Solids Struct 39:4053–4071

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