identification of elastic-plastic mechanical properties forbimetallic sheetsbyhybrid-inverse...

8/14/2019 IDENTIFICATION OF ELASTIC-PLASTIC MECHANICAL PROPERTIES FORBIMETALLIC SHEETSBYHYBRID-INVERSE APPRO

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Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 23, No. 1, February, 2010 ISSN 0894-9166

IDENTIFICATION OF ELASTIC-PLASTICMECHANICAL PROPERTIES FOR BIMETALLIC

SHEETS BY HYBRID-INVERSE APPROACH

Honglei Zhang1 Xuehui Lin1,2 Yanqun Wang1 Qian Zhang1 Yilan Kang1

(1School of Mechanical Engineering, Tianjin University, Tianjin300072, China)

(2School of Mechanical Engineering, Fuzhou University, Fuzhou350108, China)

Received 13 October 2008; revision received 9 October 2009

ABSTRACT Analysis, evaluation and interpretation of measured signals become important com-ponents in engineering research and practice, especially for material characteristic parameterswhich can not be obtained directly by experimental measurements. The present paper proposesa hybrid-inverse analysis method for the identification of the nonlinear material parameters ofany individual component from the mechanical responses of a global composite. The method cou-ples experimental approach, numerical simulation with inverse search method. The experimentalapproach is used to provide basic data. Then parameter identification and numerical simulationare utilized to identify elasto-plastic material properties by the experimental data obtained andinverse searching algorithm. A numerical example of a stainless steel clad copper sheet is consid-ered to verify and show the applicability of the proposed hybrid-inverse method. In this example,a set of material parameters in an elasto-plastic constitutive model have been identified by usingthe obtained experimental data.

KEY WORDS identification of parameters, hybrid-inverse approach, elasto-plastic mechanical

properties of bimetallic sheets

I. INTRODUCTIONWith the rapid developments in the field of material science, various advanced materials have been

constantly fabricated. Those materials are usually composed of several different components and oftenexhibit some new macro mechanical properties. If the determination of the mechanical properties ofthe individual components from the mechanical responses of the global composite becomes possible,it would be helpful in designing new composites and structures with multiple components. However,most results about internal mechanical stresses and parameters, especially some material characteristicparameters, cannot be obtained directly by experimental measurements. Some studies on the parameteridentification of composites by solving certain static or dynamic inverse problems have been reported[1,2].For example, Chen et al.[3] identified initial imperfection and strength parameters using a probabilistic

progressive failure analyzing method. Ren et al.[4]

evaluated the effect of damage nucleation parametersand pre-existing crack size on the failure stress and strain. Based on the measured real time signals of birdstrike experiment and finite element numerical solutions, the structural parameters, training efficiencyand inverse precision of the network were studied[5]. Magorou et al.[6] carried out the simultaneous

Corresponding author. Email:tju [email protected] Project supported by the National Natural Science Foundation of China (Nos. 10732080 and 10572102) and National

Basic Research Program of China (No. 2007CB714000).


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30 ACTA MECHANICA SOLIDA SINICA 2010

identification of the bending/torsion rigidities by the resolution of an inverse problem. For identifying theparameters of inelastic constitutiveequations, a method based on an evolutionary algorithm was proposedin Ref.[7]. Lin et al.[8] investigated the nonlinear damage behavior for the interfacial phase in a metalmatrix composite by using a hybrid mythology of identification. Using a combined genetic algorithm andthe nonlinear least squares method, elastic constants of composite and functional gradient plates weredetermined[9,10] in accordance with their displacement values under dynamic loads. However, all theseworks took the composites as homogenous materials and focused only on the global linear parameters ofthe composites. In fact, a great number of composites possess nonlinear mechanical properties and theinverse identification of nonlinear mechanical properties of materials is more complicated than that oflinear ones. Works of inverse identification of nonlinear parameters of homogenous materials have alsobeen reported. A unified approach for parameter identification of inelastic material models in the frameof finite element method was presented in Ref.[11]. Isotropic and anisotropic plastic parameters wererespectively identified in Refs.[12,13] by means of inhomogeneous tensile tests. An inverse approachfor the study of the through-the-thickness variation of the plastic properties of a steel structure dueto a heat treatment is proposed in Ref.[14]. Qu et al.[15] obtained viscoplastic parameters using animproved uniform random sampling method and a hybrid global optimization method. Wang et al.[16]

identified interfacial mechanical properties of the adhesive bonded interface which are time-dependentin most engineering structures due to a novel hybrid/inverse identification method. Springmann andKuna[17] presented a method for the identification of material parameters of inelastic deformation lawsby using gradient based optimization procedures. Corigliano and Mariani[18] identified the parametersof a time-dependent elastic-damage interface model for the simulation of debonding in composites. The

cyclic elastic-plastic parameters of sheet metals were identified from bending tests in Refs.[19,20]. Basedon the Gurson-Tvergaard-Needleman model, Springmann and Kuna[21] investigated the identificationof the material parameters for the ductile structural steel.

The inverse study for the determination of nonlinear properties of components in composites fromthe global nonlinear responses are more difficult due to the nonlinearity of materials as well as theincreasing number of unknown parameters to be identified. The main goal of this investigation isto propose a new hybrid/inverse procedure for the identification of nonlinear mechanical propertiesof the individual component in composite materials. The inverse engineering problem for materialcharacterization identification is solved by combining the experimental data, the mechanical model, thenumerical calculation and the identification method. As an example, a set of elastic-plastic materialparameters for a stainless steel clad copper sheet has been identified by this method. In addition, theidentification results are verified and assessed by comparing with other independent experiments.

II. SCHEME OF HYBRID-INVERSE IDENTIFICATION OF

NONLINEAR MECHANICAL PROPERTIES FOR BIMETALLIC SHEETSIt is recognized that most physical quantities such as internal mechanical stresses and material

characteristic parameters cannot be obtained directly by experimental measurements. Those finallywanted quantities are then to be evaluatedon the basis of measured deformations.This inevitably leads toa hybrid-inverse analysis, the solution of which demands appropriate mathematical/numerical algorithmscombining with the experimental results. This paper attempts to determine the nonlinear mechanicalproperties of individual components in a bimetallic composite sheet from the global mechanical responsesobtained through experiment measurement. It addresses the following aspects:

(i) A set of parameters are employed to characterize the nonlinear mechanical properties of individualcomponent metals in the bimetal sheet and thus a hybrid-inverse analysis of the nonlinear properties

of the bimetal sheet are converted into a multi-parameter inverse identification problem based on themechanical responses from experiments.

(ii) Experimentation design. In order to determine the stress-strain response for the bimetallic sheet,at least two different types of experimental data of the mechanical response are required. In this research,pure bending and tensile experiment are used.

(iii) Forward calculation. The conventional Mises yield function and the associated flow rule is usedto describe the nonlinear mechanical behavior of metal materials.


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Vol. 23, No. 1 Honglei Zhang et al.: Identification of Elastic-plastic Mechanical Properties 31

(iv) Minimization of the objective function is constructed, which represents the difference betweenthe experimental results and the results of numerical simulation. Thus the hybrid-inverse identificationof elastic-plastic parameters is transformed into an optimization problem of minimizing the objectivefunction. A gradient-based method is employed to minimize the ob jective function.

The procedure of the present identification problem is schematically illustrated in Fig.1.

Fig. 1. Scheme of the material parameter identification for a bimetaalic sheet.

2.1. Experimentation

In our experiments, the specimen used is a copper (type-T2) clad a stainless steel (type-304) sheet byexplosive bonding and its dimensions are shown in Fig.2. In the pure bending test, a four-point bendingtest is carried out, as plotted in Fig.3. In the test, the length between B and C is set to 110 mm and dis35 mm, so the momentMis uniform inz direction and of value . Furthermore, based on the assumptionthat the cross section remains plane during the test, the curvature of the specimen is determined fromthe surface strains measured by strain gauges bonded on both surfaces of the specimen and it is givenby, where is the curvature,l and u denote the lower and upper surface strains respectively and h is

Fig. 2. Copper clad stainless steel specimen used in theexperiments.

Fig. 3. Schematic illustration of experimental setup for

four-point bending test.


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the thickness of the specimen. The second type of experiment (uniaxial tension) produces the tensileload vs. strain curve.

2.2. Elasto-plastic Constitutive Model and Parameters to Be Identified

In order to describe the stress-strain response of metals under monotonic increasing load, the strainincrement d is decomposed into elastic and plastic components, del and dpl, as d = del + dpl,and the Mises yield condition for the isotropic hardening is q= 0, where0 is the yield stress and isdefined as a function of equivalent plastic strain. The yield stress can be well determined if a plot of theuniaxial-stress vs. strain is available. In the present work, the uniaxial nonlinear stress-strain behaviorof the two metals is assumed to follow the Ramberg-Osgood (R-O) relation[22]

(Y /E)=

Y +

Y

n(1)

whereEandYare Youngs modulus and the initial yield stress, respectively, and andn two strainhardening parameters for the R-O curve. Then, the plastic strain is related to the stress according tothe above equation as

pl =

E

Y

n1(2)

The above constitutive model incorporates five material parameters: two elastic constants E(Youngsmodulus) and(Poissons ratio); the initial yield stress Y ; and two R-O parametersandn for theisotropic hardening rule. Base on the above constitutive model, the finite element numerical calculation

is employed to simulate the uniaxial tension and pure bending test of the bimetal sheet by softwareABAQUS. The large-displacement formulation is used in the calculation of geometrically nonlinearbehavior expected[23].

III. HYBRID-INVERSE IDENTIFICATION FOR

MATERIAL PARAMETERS BASED ON EXPERIMENTATIONInII above, the load-strain and the moment-curvature curves are obtained for the bimetallic sheet

from the uniaxial tension and four-point bending test respectively. In the following, the elastic-plasticparameters of the component metals in the bimetallic sheet are identified by means of combiningexperimental results with the numerical calculation through an inverse approach.

Let us consider the material parameters in a constitutive model to be identified as components ofthe vector x RN. Then the inverse problem of material parameter identification can be solved by an

optimization approach and is formulated as follows: Find the vectorx

that minimizes the objectivefunction

J(x) =L

=1

J(x), Ai xi Bi (i= 1, 2, , N) (3)

where L is the total number of individual specific response quantities (denoted by ) which can bemeasured in the course of experiments and then obtained as a result of the numerical simulation. J(x)is the dimensionless function:

J(x) = 1

S

Ss=1

[Rs R (x, s)]

[Rs ]2

2

(4)

which measures the deviation between the computed individual response and the observed one fromthe experiment in which the notation denotes a parameter that defines the history of the process

in the course of the experiment, S is the number of data points, s ( = 1,...,L; s = 1,...,S) is thediscrete values of for S-th data point, Rs is the value of the -th measured response quantitycorresponding to the value of the experiment history parameters, R

(x, s) is the value of the sameresponse quantity obtained from the numerical simulation. is the weight coefficient which determinesthe relative contribution of information yielded by the -th set of experimental data, Ai, Bi are theside constraints, stipulated by some additional physical considerations, which define the search regionin the spaceRN of optimization parameters.


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The unknown material parameters of the individual component metals in the bimetallic sheet areidentified simultaneously in this work. The identification of the above material parameters, excludingPoissons ratio , which was found to be 0.28 and 0.35 for the stainless steel and the copper layersrespectively from the conventional measurements, is performed using the tension (tensile load vs. strain:P vs.) curve (= 1) and the bending (bending momentvs. curvature:M vs.) curve (= 2) whichare regarded as individual response quantities. In order to depress the ill poseness of the present inverseproblem as well as to reduce the computational expenses, the elastic-plastic parameters are identified intwo steps. As for the two Youngs module, they could be first identified directly from elastic parts ofP-andM- relationships. On the second step, the individual three plastic parameters of the two metalsare identified from the nonlinear parts of the two curves. The identification process for the elastic-plasticmodel is described as follows:

(i) in the identification of elastic parameters, the optimization variable are the Youngs mod-ule for the two layers: [Estainless steel, Ecopper]; in the identification of plastic parameters, the op-timization variables x = {x1, x2, , x6} are the plastic material parameters for the two layers:[(Y , , n)stainless steel, (Y , , n)copper],

(ii) the set of values ofRs are correlated to the set of values of the tensile loadRs =P (for = 1)

and experimental bending moment Rs = M (for = 2). In the identification of elastic parametersboth ofEstainless steel andEcopper are found from the linear part of the two experimental curves whilein the identification of plastic material parameters, both sets of values are found from the nonlinearpart of the two curves,

(iii) the function R (x, s) are obtained from the calculated tensile load (for = 1), and bending

moment (for = 2),(iv) the experiment history parameters 1s is the strain s in uniaxial tension for = 1; and

2s is

the curvatures for = 2, and the index is 1 for the uniaxial tension, 2 for the four-point bendingin Eqs.(3)-(4),

(v) all the response quantities were considered equally weighted in the formulation of the objectivefunctionJ(x).

Gradient-based optimization methods are generally effective compared with direct search methodsfor inverse problems that include relatively more parameters to be identified. The specific form of thegradient-based method, employed in the present hybrid-inverse analysis, is based on the quasi-Newtonalgorithm with updating formula which builds up an approximation of the inverse of the Hessian matrixusing the objective function values and its derivatives[24]. The derivatives of the objective function areapproximated using finite difference since their exact expressions are not explicitly known. It should benoted that difficulty may arise in the minimization of the objective functionJ(x) in the identification

of plastic parameters because of the different physical meanings and dimensions of the components ofthe vector x. The difficulty can be raveled out by normalizing the problem. We first define the initialvalues of the Nmaterial parameters as x0 = {x01, , x

0N} and then the parameter vector x can be

transform into a dimensionless vectorx by setting x = D1x, D being the diagonal N N matrixformed by the previous initial values. A new objective function Jis then defined by setting

J(x) =J(x) (5)

and the gradient ofJ is in the form

J(x) = DJ(x) (6)

IV. RESULTS AND DISCUSSIONThe results of material parameters identified for both component metals are listed in Table 1. In

order to check the accuracy of the hybrid-inverse procedure by these identified results, uniaxial tensiletests are performed on each metal layers taken from the bimetallic sheet by wire cutting processing,and the experimental stress-stain curves for both the stainless steel and the copper metals are obtained.The calculated stress-strain curves for the individual component metals via the material parametersidentified above are compared with the experimental stress-strain curves in Fig.4. It should be notedthat the results calculated with the identified parameters generally agree with those obtained from theexperiments, though a certain discrepancy in the two results exists. There might be two main reasons


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for such a discrepancy between the experimental stress-strain curves and the calculated results shownin Figs.4. It is well known that the residual stress can be inevitably induced in each metal during theexplosive bonding process and would affect the experimental results. Another possible reason for thediscrepancy would be the less flexibility of the three-parameter R-O model in describing the nonlinearbehavior of the two metals.

Table 1. Identified material parameters in the elasto-plastic model for the bimetallic sheet

Youngs modulus Initial yield stress R-O parameter R-O parameter

E (GPa) Y (MPa) nStainless steel 187.3 284.5 0.0416 6.11

Copper 115.1 134.3 0.0221 15.42

Fig. 4. Comparisons of the stress and strain curvesin uniaxial tension test and the numerical simulation

results by the constitutive models parameters for thestainless steel (SS) and aluminum (Copper) layers in

the bimetallic sheet.

Fig. 5. Comparisons of the moment-curvature curves inthe bending test and the numerical simulation by theconstitutive models parameters for bimetallic sheet (Itis a reversely placed with experiment in Fig.3).

For further check to the procedure and results in Table 1, another four-point bending test is carriedout on a bimetallic specimen. In this experiment, the specimen is reversely placed on the testing machinewith the stainless steel layer laid as the upper layer (a reversely placed with experiment in Fig.3). Figure5 shows the numerical simulation for load-curvature during bending together with the correspondingexperimental results. We can see that the simulated results agree well with those obtained in experiments.Hence, it indicates that the accuracy of the identified elastic-plastic parameters is acceptable and theseidentified parameters can be used for the prediction of the nonlinear mechanical responses of a bimetallicsheet under monotonically increasing load.

V. CONCLUSIONSThe present paper proposes a hybrid-inverse procedure to the identification of the nonlinear mate-

rial parameters of the individual component layers in a bimetallic sheet using the experimental datafrom a whole bimetallic sheet. The problem is addressed by formulating the inverse characterizationof nonlinear properties of composites as an optimization problem of minimizing ob jective functions. A

gradient-based optimization method is used to solve the problem and the nonlinear mechanical proper-ties of the individual component layers of the bimetallic sheet are identified simultaneously. The obtainednumerical results indicate that the proposed hybrid-inverse procedure is suitable for identifying mater-ial parameters and can achieveacceptable accuracy for the elastic-plasticproblems of bimetal composites.

AcknowledgementsThe authors gratefully acknowledge Prof. Karl-Hans Laermann at Bergische Uni-versity Wuppertal in Germany, for his useful help in the hybrid-inverse method.


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