- Email: [email protected]
Identification of elastic-plastic mechanical properties for bimetallic sheets by hybrid-inverse approach
Acta Mechanica Solida Sinica, Vol. 23, No. 1, February, 2010 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
IDENTIFICATION OF ELASTIC-PLASTIC MECHANICAL PROPERTIES FOR BIMETALLIC SHEETS BY HYBRID-INVERSE APPROACH Honglei Zhang1
Xuehui Lin1,2
Yanqun Wang1
Qian Zhang1
Yilan Kang1
1
( School of Mechanical Engineering, Tianjin University, Tianjin 300072, China) (2 School of Mechanical Engineering, Fuzhou University, Fuzhou 350108, China)
Received 13 October 2008; revision received 9 October 2009
ABSTRACT Analysis, evaluation and interpretation of measured signals become important components in engineering research and practice, especially for material characteristic parameters which can not be obtained directly by experimental measurements. The present paper proposes a hybrid-inverse analysis method for the identification of the nonlinear material parameters of any individual component from the mechanical responses of a global composite. The method couples experimental approach, numerical simulation with inverse search method. The experimental approach is used to provide basic data. Then parameter identification and numerical simulation are utilized to identify elasto-plastic material properties by the experimental data obtained and inverse searching algorithm. A numerical example of a stainless steel clad copper sheet is considered 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 using the obtained experimental data.
KEY WORDS identification of parameters, hybrid-inverse approach, elasto-plastic mechanical properties of bimetallic sheets
I. INTRODUCTION With 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 often exhibit some new macro mechanical properties. If the determination of the mechanical properties of the 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 characteristic parameters, cannot be obtained directly by experimental measurements. Some studies on the parameter identification 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 parameters and pre-existing crack size on the failure stress and strain. Based on the measured real time signals of bird strike experiment and finite element numerical solutions, the structural parameters, training efficiency and 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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identification of the bending/torsion rigidities by the resolution of an inverse problem. For identifying the parameters of inelastic constitutive equations, a method based on an evolutionary algorithm was proposed in Ref.[7]. Lin et al.[8] investigated the nonlinear damage behavior for the interfacial phase in a metal matrix composite by using a hybrid mythology of identification. Using a combined genetic algorithm and the nonlinear least squares method, elastic constants of composite and functional gradient plates were determined[9, 10] in accordance with their displacement values under dynamic loads. However, all these works took the composites as homogenous materials and focused only on the global linear parameters of the composites. In fact, a great number of composites possess nonlinear mechanical properties and the inverse identification of nonlinear mechanical properties of materials is more complicated than that of linear ones. Works of inverse identification of nonlinear parameters of homogenous materials have also been reported. A unified approach for parameter identification of inelastic material models in the frame of finite element method was presented in Ref.[11]. Isotropic and anisotropic plastic parameters were respectively identified in Refs.[12,13] by means of inhomogeneous tensile tests. An inverse approach for the study of the through-the-thickness variation of the plastic properties of a steel structure due to a heat treatment is proposed in Ref.[14]. Qu et al.[15] obtained viscoplastic parameters using an improved 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-dependent in most engineering structures due to a novel hybrid/inverse identification method. Springmann and Kuna[17] presented a method for the identification of material parameters of inelastic deformation laws by using gradient based optimization procedures. Corigliano and Mariani[18] identified the parameters of 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]. Based on the Gurson-Tvergaard-Needleman model, Springmann and Kuna[21] investigated the identification of the material parameters for the ductile structural steel. The inverse study for the determination of nonlinear properties of components in composites from the global nonlinear responses are more difficult due to the nonlinearity of materials as well as the increasing number of unknown parameters to be identified. The main goal of this investigation is to propose a new hybrid/inverse procedure for the identification of nonlinear mechanical properties of the individual component in composite materials. The inverse engineering problem for material characterization identification is solved by combining the experimental data, the mechanical model, the numerical calculation and the identification method. As an example, a set of elastic-plastic material parameters for a stainless steel clad copper sheet has been identified by this method. In addition, the identification results are verified and assessed by comparing with other independent experiments.
II. SCHEME OF HYBRID-INVERSE IDENTIFICATION OF NONLINEAR MECHANICAL PROPERTIES FOR BIMETALLIC SHEETS It is recognized that most physical quantities such as internal mechanical stresses and material characteristic parameters cannot be obtained directly by experimental measurements. Those finally wanted quantities are then to be evaluated on the basis of measured deformations. This inevitably leads to a hybrid-inverse analysis, the solution of which demands appropriate mathematical/numerical algorithms combining with the experimental results. This paper attempts to determine the nonlinear mechanical properties of individual components in a bimetallic composite sheet from the global mechanical responses obtained through experiment measurement. It addresses the following aspects: (i) A set of parameters are employed to characterize the nonlinear mechanical properties of individual component 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 the mechanical 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 used to describe the nonlinear mechanical behavior of metal materials.
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(iv) Minimization of the objective function is constructed, which represents the difference between the experimental results and the results of numerical simulation. Thus the hybrid-inverse identification of elastic-plastic parameters is transformed into an optimization problem of minimizing the objective function. A gradient-based method is employed to minimize the objective 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 by explosive bonding and its dimensions are shown in Fig.2. In the pure bending test, a four-point bending test is carried out, as plotted in Fig.3. In the test, the length between B and C is set to 110 mm and d is 35 mm, so the moment M is uniform in z direction and of value . Furthermore, based on the assumption that the cross section remains plane during the test, the curvature of the specimen is determined from the surface strains measured by strain gauges bonded on both surfaces of the specimen and it is given by, where κ is the curvature, εl and εu denote the lower and upper surface strains respectively and h is
Fig. 2. Copper clad stainless steel specimen used in the experiments. 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 tensile load 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 strain increment dε is decomposed into elastic and plastic components, dεel and dεpl , as dε = dεel + dεpl , and the Mises yield condition for the isotropic hardening is q = σ 0 , where σ 0 is the yield stress and is defined as a function of equivalent plastic strain. The yield stress can be well determined if a plot of the uniaxial-stress vs. strain is available. In the present work, the uniaxial nonlinear stress-strain behavior of the two metals is assumed to follow the Ramberg-Osgood (R-O) relation[22] σ n ε σ = +β (1) (Y /E) Y Y where E and Y are Young’s modulus and the initial yield stress, respectively, and β and n two strain hardening parameters for the R-O curve. Then, the plastic strain is related to the stress according to the above equation as σ σ n−1 εpl = β (2) E Y The above constitutive model incorporates five material parameters: two elastic constants E (Young’s modulus) and ν (Poisson’s ratio); the initial yield stress Y ; and two R-O parameters β and n for the isotropic 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 software ABAQUS. The large-displacement formulation is used in the calculation of geometrically nonlinear behavior expected[23] .
III. HYBRID-INVERSE IDENTIFICATION FOR MATERIAL PARAMETERS BASED ON EXPERIMENTATION In §II 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-plastic parameters of the component metals in the bimetallic sheet are identified by means of combining experimental results with the numerical calculation through an inverse approach. Let us consider the material parameters in a constitutive model to be identified as components of the 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 vector x∗ that minimizes the objective function L J α (x) = θα J α (x), Ai ≤ xi ≤ Bi (i = 1, 2, · · · , N ) (3) α=1
where L is the total number of individual specific response quantities (denoted by α) which can be measured in the course of experiments and then obtained as a result of the numerical simulation. J α (x) is the dimensionless function: J α (x) =
Sα 2 [Rsα − Rα (x, τsα )] 1 2 Sα s=1 [Rsα ]
(4)
which measures the deviation between the computed individual response and the observed one from the experiment in which the notation τ α denotes a parameter that defines the history of the process in the course of the experiment, S is the number of data points, τsα (α = 1, ..., L; s = 1, ..., S) is the discrete values of τ α for S-th data point, Rsα is the value of the α-th measured response quantity corresponding to the value of the experiment history parameter τsα , Rα (x, τsα ) is the value of the same response quantity obtained from the numerical simulation. θα is the weight coefficient which determines the relative contribution of information yielded by the α-th set of experimental data, Ai , Bi are the side constraints, stipulated by some additional physical considerations, which define the search region in the space RN of optimization parameters.
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The unknown material parameters of the individual component metals in the bimetallic sheet are identified simultaneously in this work. The identification of the above material parameters, excluding Poisson’s ratio ν, which was found to be 0.28 and 0.35 for the stainless steel and the copper layers respectively from the conventional measurements, is performed using the tension (tensile load vs. strain: P vs. ε) curve (α = 1) and the bending (bending moment vs. curvature: M vs. κ) curve (α = 2) which are regarded as individual response quantities. In order to depress the ill poseness of the present inverse problem as well as to reduce the computational expenses, the elastic-plastic parameters are identified in two steps. As for the two Young’s module, they could be first identified directly from elastic parts of P -ε and M -κ relationships. On the second step, the individual three plastic parameters of the two metals are identified from the nonlinear parts of the two curves. The identification process for the elastic-plastic model is described as follows: (i) in the identification of elastic parameters, the optimization variable are the Young’s module for the two layers: [Estainless steel , Ecopper]; in the identification of plastic parameters, the optimization variables x = {x1 , x2 , · · · , x6 } are the plastic material parameters for the two layers: [(Y, β, n)stainless steel , (Y, β, n)copper ], (ii) the set of values of Rsα are correlated to the set of values of the tensile load Rsα = P (for α = 1) and experimental bending moment Rsα = M (for α = 2). In the identification of elastic parameters both of Estainless steel and Ecopper are found from the linear part of the two experimental curves while in the identification of plastic material parameters, both sets of values are found from the nonlinear part of the two curves, (iii) the function Rα (x, τsα ) are obtained from the calculated tensile load (for α = 1), and bending moment (for α = 2), 2 (iv) the experiment history parameters τs1 is the strain εα s in uniaxial tension for α = 1; and τs is α the curvature κs for α = 2, and the index α is 1 for the uniaxial tension, 2 for the four-point bending in Eqs.(3)-(4), (v) all the response quantities were considered equally weighted in the formulation of the objective function J(x). Gradient-based optimization methods are generally effective compared with direct search methods for inverse problems that include relatively more parameters to be identified. The specific form of the gradient-based method, employed in the present hybrid-inverse analysis, is based on the quasi-Newton algorithm with updating formula which builds up an approximation of the inverse of the Hessian matrix using the objective function values and its derivatives[24]. The derivatives of the objective function are approximated using finite difference since their exact expressions are not explicitly known. It should be noted that difficulty may arise in the minimization of the objective function J(x) in the identification of plastic parameters because of the different physical meanings and dimensions of the components of the vector x. The difficulty can be raveled out by normalizing the problem. We first define the initial values of the N material parameters as x0 = {x01 , · · · , x0N } and then the parameter vector x can be ˜ by setting x ˜ = D −1 x, D being the diagonal N × N matrix transform into a dimensionless vector x formed by the previous initial values. A new objective function J˜ is then defined by setting
and the gradient of J˜ is in the form
˜ x) = J(x) J(˜
(5)
˜ x) = D∇J(x) ∇J(˜
(6)
IV. RESULTS AND DISCUSSION The 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 tensile tests 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 parameters identified above are compared with the experimental stress-strain curves in Fig.4. It should be noted that the results calculated with the identified parameters generally agree with those obtained from the experiments, 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 shown in Figs.4. It is well known that the residual stress can be inevitably induced in each metal during the explosive bonding process and would affect the experimental results. Another possible reason for the discrepancy would be the less flexibility of the three-parameter R-O model in describing the nonlinear behavior of the two metals. Table 1. Identified material parameters in the elasto-plastic model for the bimetallic sheet
Stainless steel Copper
Young’s modulus E (GPa) 187.3 115.1
Initial yield stress Y (MPa) 284.5 134.3
Fig. 4. Comparisons of the stress and strain curves in uniaxial tension test and the numerical simulation results by the constitutive models parameters for the stainless steel (SS) and aluminum (Copper) layers in the bimetallic sheet.
R-O parameter β 0.0416 0.0221
R-O parameter n 6.11 15.42
Fig. 5. Comparisons of the moment-curvature curves in the bending test and the numerical simulation by the constitutive models parameters for bimetallic sheet (It is 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 carried out on a bimetallic specimen. In this experiment, the specimen is reversely placed on the testing machine with the stainless steel layer laid as the upper layer (a reversely placed with experiment in Fig.3). Figure 5 shows the numerical simulation for load-curvature during bending together with the corresponding experimental 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 these identified parameters can be used for the prediction of the nonlinear mechanical responses of a bimetallic sheet under monotonically increasing load.
V. CONCLUSIONS The present paper proposes a hybrid-inverse procedure to the identification of the nonlinear material parameters of the individual component layers in a bimetallic sheet using the experimental data from a whole bimetallic sheet. The problem is addressed by formulating the inverse characterization of nonlinear properties of composites as an optimization problem of minimizing objective functions. A gradient-based optimization method is used to solve the problem and the nonlinear mechanical properties of the individual component layers of the bimetallic sheet are identified simultaneously. The obtained numerical results indicate that the proposed hybrid-inverse procedure is suitable for identifying material parameters and can achieve acceptable accuracy for the elastic-plastic problems of bimetal composites. Acknowledgements The authors gratefully acknowledge Prof. Karl-Hans Laermann at Bergische University Wuppertal in Germany, for his useful help in the hybrid-inverse method.
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References [1] Laermann,K.H., Numerical methods for interpretation of measured data in solid mechanics. Proc. 21st Symposium on Experimental Mechanics of Solids, Jachranka, Poland, 2004a: 45-54. [2] Laermann,K.H., Experimental/Numerical Parameter Identification for Visco-elastic Problems. The Archive of Mechanical Engineering, 2004b, LI(4): 581-594. [3] Chen,J.Q., Wang,X.Y. and Luo,C., Reliability analysis of frp laminated plates with consideration of both initial imperfection and failure sequence. Acta Mechanica Solida Sinica, 2002, 15(2): 227-235. [4] Ren,H.L. and Ning,J.G., A dynamic compressive damage model for ceramics. Acta Mechanica Solida Sinica, 2006, 27(3): 303-306 (in Chinese). [5] Bai,J.Z. and Sun,Q., Neural-Network Based Bird Strike Loading Inverse to Aircraft Windshield. Acta Mechanica Solida Sinica, 2005, 26(1): 77-82 (in Chinese). [6] L.Le Magoroua, F.Bosa and F.Rouger, Identification of constitutive laws for wood-based panels by means of an inverse method. Composites Science and Technology, 2002, 62: 591-596. [7] Tomonari Furukawa and Genki Yagawa, Inelastic constitutive parameter identification using an evolutionary algorithm with continuous individuals. International Journal for Numerical Methods in Engineering, 1997, 40: 1071-1090. [8] Lin,X.H., Kang,Y.L., Qin,Q.H. and Fu,D.H., Identification of interfacial parameters in a particle reinforced metal matrix composite Al6061-10%Al2 O3 by hybrid method and genetic algorithm. Computational Materials Science, 2005, 32: 47-56. [9] Liu,G.R. and Han,X., An inverse procedure for determination of material property of FGM. International Symposium on Inverse Problems in Engineering Mechanics, Nagano City, Japan: 2001, 63-71. [10] Liu,G.R., Han,X. and Lam,K.Y., A combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves. Computer Methods in Applied Mechanics and Engineering, 2002, 191: 1909-1916. [11] Mahnken,R. and Stein,E., A unified approach for parameter identification of inelastic material models in the frame of the finite element method. Computer Methods in Applied Mechanics and Engineering, 1996, 136: 225-258. [12] Khalfallah,A., Bel Hadj Salah and H.,Dogui,A., Anisotropic parameter identification using inhomogeneous tensile test. European Journal of Mechanics, A/Solids, 2002, 21: 927-942. [13] Meuwissen,M.H.H., Oomens,C.W.J., Baaijens,F.P.T., Petterson,R. and Janssen,J.D., Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method. J. Mat. Processing Tech, 1998, 75: 204-211. [14] Niclas,A. and Bourgeois,L. An inverse approach to determine the non-linear properties of induction heattreated steels. European Journal of Mechanics, A/Solids, 2000, 19: 69-88. [15] Qu,J., Jin,Q.L. and Xu,B.Y., Parameter identification for improved viscoplastic model considering dynamic recrystallization. International Journal of Plasticity, 2005, 21: 1267-1302. [16] Wang,J., Kang,Y.L., Qin,Q.H., Fu,D.H. and Li,X.Q., Identification of time-dependent interfacial mechanical properties of adhesive by hybrid/inverse method. Computational Materials Science, 2008, 43: 1160-1164. [17] Springmann,M. and Kuna,M. Identification of material parameters of the Rousselier model by non-linear optimization. Computational Material Science, 2003, 26: 202-209. [18] Alberto,Corigliano and Stefano,Mariani, Parameter identification of a time-dependent elastic-damage interface model for the simulation of debonding in composites. Composites Science and Technology, 2001, 61: 191-203. [19] Yoshida,U.M and Toropov,V.V., Identification of material parameters in constitutive model for sheet metals from cyclic bending tests. International Journal of Mechanical Science, 1998, 40: 237-249. [20] Zhao,K.M. and Lee,J.K., Generation of cyclic stress-strain curves for sheet metals. Journal of Engineering Mateials and Technology, 2001, 123: 390-397. [21] M.Springmann and M.Kuna, Identification of material parameters of the Gurson-Tvergaard-Needleman model by combined experimental and numerical techniques. Computational Materials Science, 2005, 33: 544-552. [22] Ramberg,W. and Osgood,W.R., Description of stress-strain curve by three parameters. J. NACA Technical Note, 1943, 902: 13. [23] ABAQUS/Standard User’s Manual for Version 6.2, vols. I, II and III, Hibbitt, Karlsson & Sorensen, Inc. [24] Gill,P.E., Murray,W. and Wright,M.H., Practical Optimization. Academic Press, 1982.
ISSN 0894-9166
IDENTIFICATION OF ELASTIC-PLASTIC MECHANICAL PROPERTIES FOR BIMETALLIC SHEETS BY HYBRID-INVERSE APPROACH Honglei Zhang1
Xuehui Lin1,2
Yanqun Wang1
Qian Zhang1
Yilan Kang1
1
( School of Mechanical Engineering, Tianjin University, Tianjin 300072, China) (2 School of Mechanical Engineering, Fuzhou University, Fuzhou 350108, China)
Received 13 October 2008; revision received 9 October 2009
ABSTRACT Analysis, evaluation and interpretation of measured signals become important components in engineering research and practice, especially for material characteristic parameters which can not be obtained directly by experimental measurements. The present paper proposes a hybrid-inverse analysis method for the identification of the nonlinear material parameters of any individual component from the mechanical responses of a global composite. The method couples experimental approach, numerical simulation with inverse search method. The experimental approach is used to provide basic data. Then parameter identification and numerical simulation are utilized to identify elasto-plastic material properties by the experimental data obtained and inverse searching algorithm. A numerical example of a stainless steel clad copper sheet is considered 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 using the obtained experimental data.
KEY WORDS identification of parameters, hybrid-inverse approach, elasto-plastic mechanical properties of bimetallic sheets
I. INTRODUCTION With 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 often exhibit some new macro mechanical properties. If the determination of the mechanical properties of the 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 characteristic parameters, cannot be obtained directly by experimental measurements. Some studies on the parameter identification 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 parameters and pre-existing crack size on the failure stress and strain. Based on the measured real time signals of bird strike experiment and finite element numerical solutions, the structural parameters, training efficiency and 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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identification of the bending/torsion rigidities by the resolution of an inverse problem. For identifying the parameters of inelastic constitutive equations, a method based on an evolutionary algorithm was proposed in Ref.[7]. Lin et al.[8] investigated the nonlinear damage behavior for the interfacial phase in a metal matrix composite by using a hybrid mythology of identification. Using a combined genetic algorithm and the nonlinear least squares method, elastic constants of composite and functional gradient plates were determined[9, 10] in accordance with their displacement values under dynamic loads. However, all these works took the composites as homogenous materials and focused only on the global linear parameters of the composites. In fact, a great number of composites possess nonlinear mechanical properties and the inverse identification of nonlinear mechanical properties of materials is more complicated than that of linear ones. Works of inverse identification of nonlinear parameters of homogenous materials have also been reported. A unified approach for parameter identification of inelastic material models in the frame of finite element method was presented in Ref.[11]. Isotropic and anisotropic plastic parameters were respectively identified in Refs.[12,13] by means of inhomogeneous tensile tests. An inverse approach for the study of the through-the-thickness variation of the plastic properties of a steel structure due to a heat treatment is proposed in Ref.[14]. Qu et al.[15] obtained viscoplastic parameters using an improved 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-dependent in most engineering structures due to a novel hybrid/inverse identification method. Springmann and Kuna[17] presented a method for the identification of material parameters of inelastic deformation laws by using gradient based optimization procedures. Corigliano and Mariani[18] identified the parameters of 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]. Based on the Gurson-Tvergaard-Needleman model, Springmann and Kuna[21] investigated the identification of the material parameters for the ductile structural steel. The inverse study for the determination of nonlinear properties of components in composites from the global nonlinear responses are more difficult due to the nonlinearity of materials as well as the increasing number of unknown parameters to be identified. The main goal of this investigation is to propose a new hybrid/inverse procedure for the identification of nonlinear mechanical properties of the individual component in composite materials. The inverse engineering problem for material characterization identification is solved by combining the experimental data, the mechanical model, the numerical calculation and the identification method. As an example, a set of elastic-plastic material parameters for a stainless steel clad copper sheet has been identified by this method. In addition, the identification results are verified and assessed by comparing with other independent experiments.
II. SCHEME OF HYBRID-INVERSE IDENTIFICATION OF NONLINEAR MECHANICAL PROPERTIES FOR BIMETALLIC SHEETS It is recognized that most physical quantities such as internal mechanical stresses and material characteristic parameters cannot be obtained directly by experimental measurements. Those finally wanted quantities are then to be evaluated on the basis of measured deformations. This inevitably leads to a hybrid-inverse analysis, the solution of which demands appropriate mathematical/numerical algorithms combining with the experimental results. This paper attempts to determine the nonlinear mechanical properties of individual components in a bimetallic composite sheet from the global mechanical responses obtained through experiment measurement. It addresses the following aspects: (i) A set of parameters are employed to characterize the nonlinear mechanical properties of individual component 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 the mechanical 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 used to describe the nonlinear mechanical behavior of metal materials.
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 between the experimental results and the results of numerical simulation. Thus the hybrid-inverse identification of elastic-plastic parameters is transformed into an optimization problem of minimizing the objective function. A gradient-based method is employed to minimize the objective 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 by explosive bonding and its dimensions are shown in Fig.2. In the pure bending test, a four-point bending test is carried out, as plotted in Fig.3. In the test, the length between B and C is set to 110 mm and d is 35 mm, so the moment M is uniform in z direction and of value . Furthermore, based on the assumption that the cross section remains plane during the test, the curvature of the specimen is determined from the surface strains measured by strain gauges bonded on both surfaces of the specimen and it is given by, where κ is the curvature, εl and εu denote the lower and upper surface strains respectively and h is
Fig. 2. Copper clad stainless steel specimen used in the experiments. 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 tensile load 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 strain increment dε is decomposed into elastic and plastic components, dεel and dεpl , as dε = dεel + dεpl , and the Mises yield condition for the isotropic hardening is q = σ 0 , where σ 0 is the yield stress and is defined as a function of equivalent plastic strain. The yield stress can be well determined if a plot of the uniaxial-stress vs. strain is available. In the present work, the uniaxial nonlinear stress-strain behavior of the two metals is assumed to follow the Ramberg-Osgood (R-O) relation[22] σ n ε σ = +β (1) (Y /E) Y Y where E and Y are Young’s modulus and the initial yield stress, respectively, and β and n two strain hardening parameters for the R-O curve. Then, the plastic strain is related to the stress according to the above equation as σ σ n−1 εpl = β (2) E Y The above constitutive model incorporates five material parameters: two elastic constants E (Young’s modulus) and ν (Poisson’s ratio); the initial yield stress Y ; and two R-O parameters β and n for the isotropic 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 software ABAQUS. The large-displacement formulation is used in the calculation of geometrically nonlinear behavior expected[23] .
III. HYBRID-INVERSE IDENTIFICATION FOR MATERIAL PARAMETERS BASED ON EXPERIMENTATION In §II 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-plastic parameters of the component metals in the bimetallic sheet are identified by means of combining experimental results with the numerical calculation through an inverse approach. Let us consider the material parameters in a constitutive model to be identified as components of the 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 vector x∗ that minimizes the objective function L J α (x) = θα J α (x), Ai ≤ xi ≤ Bi (i = 1, 2, · · · , N ) (3) α=1
where L is the total number of individual specific response quantities (denoted by α) which can be measured in the course of experiments and then obtained as a result of the numerical simulation. J α (x) is the dimensionless function: J α (x) =
Sα 2 [Rsα − Rα (x, τsα )] 1 2 Sα s=1 [Rsα ]
(4)
which measures the deviation between the computed individual response and the observed one from the experiment in which the notation τ α denotes a parameter that defines the history of the process in the course of the experiment, S is the number of data points, τsα (α = 1, ..., L; s = 1, ..., S) is the discrete values of τ α for S-th data point, Rsα is the value of the α-th measured response quantity corresponding to the value of the experiment history parameter τsα , Rα (x, τsα ) is the value of the same response quantity obtained from the numerical simulation. θα is the weight coefficient which determines the relative contribution of information yielded by the α-th set of experimental data, Ai , Bi are the side constraints, stipulated by some additional physical considerations, which define the search region in the space RN of optimization parameters.
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The unknown material parameters of the individual component metals in the bimetallic sheet are identified simultaneously in this work. The identification of the above material parameters, excluding Poisson’s ratio ν, which was found to be 0.28 and 0.35 for the stainless steel and the copper layers respectively from the conventional measurements, is performed using the tension (tensile load vs. strain: P vs. ε) curve (α = 1) and the bending (bending moment vs. curvature: M vs. κ) curve (α = 2) which are regarded as individual response quantities. In order to depress the ill poseness of the present inverse problem as well as to reduce the computational expenses, the elastic-plastic parameters are identified in two steps. As for the two Young’s module, they could be first identified directly from elastic parts of P -ε and M -κ relationships. On the second step, the individual three plastic parameters of the two metals are identified from the nonlinear parts of the two curves. The identification process for the elastic-plastic model is described as follows: (i) in the identification of elastic parameters, the optimization variable are the Young’s module for the two layers: [Estainless steel , Ecopper]; in the identification of plastic parameters, the optimization variables x = {x1 , x2 , · · · , x6 } are the plastic material parameters for the two layers: [(Y, β, n)stainless steel , (Y, β, n)copper ], (ii) the set of values of Rsα are correlated to the set of values of the tensile load Rsα = P (for α = 1) and experimental bending moment Rsα = M (for α = 2). In the identification of elastic parameters both of Estainless steel and Ecopper are found from the linear part of the two experimental curves while in the identification of plastic material parameters, both sets of values are found from the nonlinear part of the two curves, (iii) the function Rα (x, τsα ) are obtained from the calculated tensile load (for α = 1), and bending moment (for α = 2), 2 (iv) the experiment history parameters τs1 is the strain εα s in uniaxial tension for α = 1; and τs is α the curvature κs for α = 2, and the index α is 1 for the uniaxial tension, 2 for the four-point bending in Eqs.(3)-(4), (v) all the response quantities were considered equally weighted in the formulation of the objective function J(x). Gradient-based optimization methods are generally effective compared with direct search methods for inverse problems that include relatively more parameters to be identified. The specific form of the gradient-based method, employed in the present hybrid-inverse analysis, is based on the quasi-Newton algorithm with updating formula which builds up an approximation of the inverse of the Hessian matrix using the objective function values and its derivatives[24]. The derivatives of the objective function are approximated using finite difference since their exact expressions are not explicitly known. It should be noted that difficulty may arise in the minimization of the objective function J(x) in the identification of plastic parameters because of the different physical meanings and dimensions of the components of the vector x. The difficulty can be raveled out by normalizing the problem. We first define the initial values of the N material parameters as x0 = {x01 , · · · , x0N } and then the parameter vector x can be ˜ by setting x ˜ = D −1 x, D being the diagonal N × N matrix transform into a dimensionless vector x formed by the previous initial values. A new objective function J˜ is then defined by setting
and the gradient of J˜ is in the form
˜ x) = J(x) J(˜
(5)
˜ x) = D∇J(x) ∇J(˜
(6)
IV. RESULTS AND DISCUSSION The 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 tensile tests 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 parameters identified above are compared with the experimental stress-strain curves in Fig.4. It should be noted that the results calculated with the identified parameters generally agree with those obtained from the experiments, though a certain discrepancy in the two results exists. There might be two main reasons
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ACTA MECHANICA SOLIDA SINICA
2010
for such a discrepancy between the experimental stress-strain curves and the calculated results shown in Figs.4. It is well known that the residual stress can be inevitably induced in each metal during the explosive bonding process and would affect the experimental results. Another possible reason for the discrepancy would be the less flexibility of the three-parameter R-O model in describing the nonlinear behavior of the two metals. Table 1. Identified material parameters in the elasto-plastic model for the bimetallic sheet
Stainless steel Copper
Young’s modulus E (GPa) 187.3 115.1
Initial yield stress Y (MPa) 284.5 134.3
Fig. 4. Comparisons of the stress and strain curves in uniaxial tension test and the numerical simulation results by the constitutive models parameters for the stainless steel (SS) and aluminum (Copper) layers in the bimetallic sheet.
R-O parameter β 0.0416 0.0221
R-O parameter n 6.11 15.42
Fig. 5. Comparisons of the moment-curvature curves in the bending test and the numerical simulation by the constitutive models parameters for bimetallic sheet (It is 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 carried out on a bimetallic specimen. In this experiment, the specimen is reversely placed on the testing machine with the stainless steel layer laid as the upper layer (a reversely placed with experiment in Fig.3). Figure 5 shows the numerical simulation for load-curvature during bending together with the corresponding experimental 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 these identified parameters can be used for the prediction of the nonlinear mechanical responses of a bimetallic sheet under monotonically increasing load.
V. CONCLUSIONS The present paper proposes a hybrid-inverse procedure to the identification of the nonlinear material parameters of the individual component layers in a bimetallic sheet using the experimental data from a whole bimetallic sheet. The problem is addressed by formulating the inverse characterization of nonlinear properties of composites as an optimization problem of minimizing objective functions. A gradient-based optimization method is used to solve the problem and the nonlinear mechanical properties of the individual component layers of the bimetallic sheet are identified simultaneously. The obtained numerical results indicate that the proposed hybrid-inverse procedure is suitable for identifying material parameters and can achieve acceptable accuracy for the elastic-plastic problems of bimetal composites. Acknowledgements The authors gratefully acknowledge Prof. Karl-Hans Laermann at Bergische University Wuppertal in Germany, for his useful help in the hybrid-inverse method.
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