- Email: [email protected]
Inverse Characterization of Material Constitutive Parameters for Dynamic Applications
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 114 (2015) 784 – 791
1st International Conference on Structural Integrity
Inverse Characterization of Material Constitutive Parameters for Dynamic Applications Inês Oliveiraa, Pedro Teixeiraa*, Fernando Ferreirab, Ana Reisa,c a
Institute of Science And Innovation in Mechanical and Industrial Engineering (INEGI), Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal b Polytechnic Institute of Porto (ISEP), Rua Dr. António Bernardino de Almeida 431, 4249-015 Porto, Portugal c Faculty of Engineering of the University of Porto (FEUP), Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Abstract Accurate modelling of mechanical dynamic behavior is a prerequisite for an effective analysis of both manufacturing processes and dynamic load during service life of the components. Consequently, precise and reliable rate dependent constitutive models, and related parameters, are required to represent the plastic behavior of metals undergoing high-strain rate deformations. However, these parameters are not always directly measurable. This study presents an inverse methodology to identify the dynamic properties of metallic materials by using the Split Hopkinson pressure bar. First, a symmetric finite element model for the simulation of SHPB test was implemented into ABAQUS/Explicit software. An inverse analysis methodology was then established, enabling the determination of the material parameters for the Johnson-Cook constitutive law. A user defined objective function was used to measure the optimality of the parameters, by evaluating the error between numerical and experimental results of SHPB test, namely the elastic strain wave’s evolution with time. The minimization of the objective function with respect to the material parameters was accomplished using the Levenberg–Marquardt optimization algorithm. ©2015 2015Published The Authors. Published by isElsevier © by Elsevier Ltd. This an open Ltd. access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering. Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering
Keywords: Hopkinson bar; High strain rate; Aluminum AA5182; Constitutive modeling; Finite element optimization
* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: [email protected]
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering
doi:10.1016/j.proeng.2015.08.027
785
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
1. Introduction Computational simulation plays an indispensable role in the integrity analysis of components and structures during their manufacture and service life. Robust numerical models are needed to shorten design cycles, reduce manufacturing costs and create reliable final products. The accuracy of a FE simulation largely depends on the reliability of the input data, namely geometry, applied load, contact properties, boundary conditions and material data. Specifically, the knowledge of the material properties is of utmost importance to perform useful simulations. A correct material model requires not only an appropriate material constitutive law but also the determination of its parameters at different conditions. In line with these trends and needs, dynamic behavior of metallic materials has become one of the most current concerns. Besides, in the most of the applications of the metallic materials, e.g. industries as aerospace, military, civil engineering and automotive, the loading during service life has dynamic characteristics. There are several methods to determine dynamic material properties [1, 2]. The split Hopkinson pressure bar (SHPB) technique has become one of the most popular experimental tools for the study of dynamic behavior of solid materials. Attempts to apply the SHPB to the determination of constitutive laws parameters are presented by several authors [1, 2]. Typically, these parameters are estimated by the curve fitting method, using the stress-strain diagram derived from the recorded strain gage signals in a SHPB experiment. The engineering stress ( V s ) and strain ( H s ) as function of transmitted pulse strain are expressed by:
Vs
E
A Ht As
(1)
Hs
2C0 t H r dt L ³0
(2)
Hs
2C0 Hr L
(3)
where, E is the modulus of elasticity of the bar material, A is the cross-sectional area of the bar, As is the crosssectional area of the specimen, L is the gauge length of the specimen and C 0 is the speed of the stress wave in the pressure bar. H r and H t are the reflected and transmitted strains in the incident and transmitted bar, respectively. As verified, this strategy is highly dependent on constants used for the stress-strain computation. To reduce the user inputs and associated errors, it is proposed, in the present work, an alternative inverse approach. It is based on the quantitative assessment of the combination of constitutive parameters that leads to a minimum error between the time dependent elastic waves, acquired experimentally, and the equivalent ones attained by process simulation. In short words, the main idea of this approach is to perform a set of experiments, simulate them, and fit the material parameters in order to numerically obtain the same results as in the experimental ones, using as comparison the reflected and transmitted strains [3]. A set of split Hopkinson pressure bar tensile tests were conducted. Specimens, made on aluminum AA5182, were subjected to different strain-rates as result of the variation of the impact pressure and the geometrical properties (as the gage length). In terms of numerical work, an optimization procedure that makes use of MATLAB Levenberg– Marquardt routine was developed. This procedure uses a script to generate the input file for ABAQUS in order to obtain finite element solution for SHPB process. After simulation another executable file is used to read desirable data from ABAQUS output database. The material constitutive law and its parameters where introduced in the numerical analysis by using a user-defined VUHARD ABAQUS Subroutine. These steps were repeated allowing the iterative updating of the material parameters in the Finite Element (FE) model. The philosophy was to minimize the difference between numerically predicted and experimentally measured strain elastic waves according to a predefined error criterion.
786
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
2. Materials and Methods 2.1. Experimental Setup The first use of a long thin bar to measure the pulse shape induced by an impact is attributed to Hopkinson (1914) [4]. Kolsky, in 1949 [5], developed an experimental setup with two long bars and a short specimen, as is presented in Fig. 1.
Fig. 1. Schematic representation of a split Hopkinson pressure-bar device.
This traditional SHPB or Kolsky apparatus, used for dynamic compression experiments, consists of a striker, incident, and transmission bar. A gas gun, whose pressure controls the velocity of the striker bar, is used to drive the striker bar towards the incident bar. The impact of the striker bar on the incident bar is responsible for the creation of an elastic stress wave, which propagates trough the incident bar towards the specimen, which is sandwiched between the incident and the transmission bar. When stress wave reaches the incident bar/specimen interface, part of the stress wave is transferred to the specimen, causing a quick deformation, while part of the stress wave is reflected back to incident bar, called reflected wave [6]. The technique has been extended to tension [7] and to torsion [8]. In this work, dynamic tensile tests were performed using a tensile split Hopkinson pressure bar, whose setup is presented in Fig. 2.
Fig. 2. SHPB experimental setup. Grips Detail.
The setup of the Hopkinson bar used in this study includes two steel bars of 16 mm diameter and a gas gun equipped with a pneumatic valve. The incident (input) bar is 5735 mm long and the strain gauge located at 2400 mm from the impact end of the bar. The transmission (output) bar has a length of 2845 mm and the stain gauge positioned at 330 mm away from the specimen. The length of the striker bar is 1900 mm. Electronic equipment allows the measurement of the impact velocity. Signal conditioning with wide band amplification is also provided by adequate equipment to measure the dynamic strain in both bars. A LeCroy 9450A scope is used to record the signal at 2 MHz sample rate. The recorded signals are acquired by a personal computer through a RS232 connection. For testing metallic materials, the dog-bone shaped specimens were connected by screws to the bars grips. This assembling strategy ensures a good transmission of the waves [6]. The tested geometries are presented in Fig. 3.
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
Fig. 3. SHPB tensile specimen’s geometries. Dimensions are in millimetres.
2.2. Simulation Tool The commercial FEA software ABAQUS was used to model the split Hopkinson pressure bar. As referred, the experimental setup is composed of a cylindrical input bar of 16 mm diameter and 5735 mm length, a flat dog-bone specimen placed between the input bar and the output bar of 16 mm diameter and 2845 mm length. For simplicity, and due to symmetry, only one quarter of the problem was modelled and appropriate boundary conditions were imposed. In the present FE model, the specimen is considered as a deformable part, meshed with C3D8R elements, which is an eight node linear brick element, with reduced integration. The thickness of the specimen is equal to 1 mm and two elements were created in this direction. The bars, also meshed using C3D8R elements, were considered deformable parts, working only in the elastic regime. A total 12202 elements were considered in the FE model. The impact simulation was performed by applying a prescribed pressure on the bar free end. This pressure was calculated based on the incident wave acquired during the experiments. Element sets were created in both input and output bar, allowing the measure of elastic displacements in the same points as in the experimental tests. The simulation process time was fixed to 270 μs in agreement with experiments. A dynamic explicit time integration scheme (ABAQUS/Explicit) was employed
Fig. 4. Symmetric finite element model of the specimen and the pressure bars.
787
788
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
2.3. Optimization Procedure An inverse iterative FE technique was used to solve the problem of identifying the parameters for the JohnsonCook constitutive law, presented below.
V eq
§
§
©
©
A BH ¨¨1 C ln ¨ HH n
0
·· ¸ ¸¸ ¹¹
(4)
The parameters were estimated by minimizing the objective function, which expresses the discrepancy between the experimentally measured and the numerically computed strains at certain points of the bars. The mathematical model for inverse analysis can be expressed as: Find: A, B, n, C, H 0 Minimize: I ( A, B, n, C, H 0 ) With:
I
§ H iexp H inum · ¨ ¸ ¦ s i 1© ¹ s
2
(5)
where ݏis the number of experimental data points, H iexp the experimentally measured data point and H inum the numerically computed strain as a function of the unknown material parameters. The optimization problem expressed above was solved by the Levenberg-Marquardt (LM) algorithm. It consists in an iterative technique that finds the minimum of a function which is expressed as the sum of squares of nonlinear functions [9, 10]. It has become a standard technique for nonlinear least-squares problems and can be thought as a combination of steepest descent and the Gauss-Newton (GN) method. The procedure consists in a sequence of linear least squares approximations to the nonlinear problem. This algorithm presents a quadratic convergence that makes it very fast. However, the results highly depend on the initial parameters. As in the most of the cases, an accurate prediction of an appropriate set of initial parameters is not easy, being unrealistic for many applications. On the other hand, gradient descent algorithm is less dependent on the initial guess, but it is a first-order optimization algorithm, meaning that its speed of convergence is generally low. Therefore, the LevenbergMarquardt algorithm is a hybrid optimization technique which makes use of the positive attributes of both. In short, when the current solution is far from the correct one, the algorithm makes use of the steepest descent method properties, guaranteeing the convergence. When the current solution is close to the optimal solution, it uses the Gauss-Newton technique. The optimization procedure, summarized in the following diagram, was scripted under MATLAB. The optimization progresses from a given vector of initial material parameters towards the final ones, by solving the direct problem, defined by the SHPB finite element model. A developed Python script generates the input for ABAQUS. The material constitutive law is introduced in the numerical simulation using a user-defined FORTRAN subroutine, called VUHARD. Another Python script is used to read and extract the strain history from the ABAQUS output file. The MATLAB program reads both simulation and experimental results, and calculates the objective function, as explained above. The optimization process, governed by Levenberg-Marquardt algorithm, will continue until the objective function reaches the minimum specified by the convergence criterion. For each iteration, Johnson-Cook constitutive law parameters are updated and written by MATLAB to serve as ABAQUS inputs for the next direct problem.
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
Fig. 5. Flow-chart of the inverse method for material parameter identification.
3. Results and Discussion 3.1. Optimization results For the optimization algorithm, an initial set of parameters was defined by using a simple preliminary approximation. This preliminary approximation follows the procedure of the traditional approach, i.e. the stressstrain curves calculated based on strain-time evolution were used to perform an initial estimative of material parameters. The calculation process was stopped after 100 iterations or after the stop criterion was attained. The results of the optimization process are presented in Fig. 6 and 7. The parameters for the Johnson-Cook constitutive law obtained by the iterative program are summarized in Table 1. The time-strain evolutions obtained by the optimization procedure show some, but non-dramatic disparity, when compared with the wave achieved from experimental results, with especially significance in the duration of the output bar wave. The usual SHPB assumptions and simplifications can be the cause for the difference. Table 1: Identified parameters for the Johnson-Cook constitutive law Parameter
Value
A B n C H0
77.57 557.23 0.53 0.00067 0.00010
789
790
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
In addition, and as is possible to confirm in the Fig. 6 and 7, one of the major issues of the SHPB is the acquisition of data containing substantial measurement noises or errors, typically associated to experimental uncontrollable factors. The noise has a negative influence on the optimization procedure. The error (difference between both curves) is calculated at every point, meaning that it is substantially affected by the curve’s quality. However, despite the verified difference, both curves present the same general appearance and maximum values.
Fig. 6. Experimental and optimized numerical results for the geometry 1.
Fig. 7. Experimental and optimized numerical results for the geometry 2.
4. Summary and Conclusions Mechanical material characterization represents a research challenge, especially in the dynamic domain, since the mechanical properties of some materials are influenced by the loading rate. The main objective of this research work is to contribute to the numerical modelling improvement in the prediction of workpiece behaviour during dynamic processes. For such purpose, high strain rate tensile tests have been performed on AA 5182 specimens, by means of
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
791
a SHPB apparatus. Afterwards, the same tests have been numerically reproduced with a finite element commercial code. A material characterization methodology to determine the material parameters of a given material constitutive model from the given high strain rate experiment was then established. The characterization approach is based on an inverse technique in which an inverse problem is formulated and solved as an optimization procedure. The input of the optimization procedure is the characteristic signal from the high strain rate experiment, in this case, the strain-time evolution. The output of the procedure is the optimum set of material parameters determined by fitting a numerical simulation to the high strain rate experimental signal. This characterization procedure demonstrates advantages in comparison with other techniques. Numerical results of the characterization process showed that the implemented algorithm is effective in determining the Johnson-Cook material constants for the aluminium alloy 5182. At this stage, some words should be dedicated to possible strategies that can be used to improve the achieved results. In first place, more experimental tests should be conducted in order to validate the numerical model results and optimization process conclusions. More geometries and strain-rates should be investigated. Additionally, the approach must be extended to different materials laws and materials. An improvement of the presented optimization procedure is required, possibly by introducing a new method to perform an initial assessment of parameters, allowing a faster convergence of the results. Acknowledgements Authors gratefully acknowledge funding of Project SAESCTN-PII&DT/1/2011 co-financed by Programa Operacional Regional do Norte (ON.2 – O Novo Norte), under Quadro de Referência Estratégico Nacional (QREN), through Fundo Europeu de Desenvolvimento Regional (FEDER). Besides, the authors also gratefully acknowledge funding of Project EXPL/EMS-TEC/2419/2013, which has the financial support of FCT, Portuguese national funding agency for science, research and technology. References [1] Sasso, M., G. Newaz, and D. Amodio, Material characterization at high strain rate by Hopkinson bar tests and finite element optimization. Materials Science and Engineering: A, 25 July 2008. 487(1-2): p. 289-300. [2] El-Magd, E. and M. Abouridouane, Characterization, modelling and simulation of deformation and fracture behaviour of the light-weight wrought alloys under high strain rate loading. International Journal of Impact Engineering, May 2006. 32(5): p. 741–758. [3] Lee, O.S. and M.S. Kim, Dynamic material property characterization by using split Hopkinson pressure bar (SHPB) technique. Nuclear Engineering and Design, December 2003. 226(2): p. 119–125. [4] Hopkinson, B., A Method of Measuring the Pressure Produced in the Detonation of High Explosives or by the Impact of Bullets. Philos. Trans. R. Soc. (London) 1914. 213: p. 437-456. [5] Kolsky, H., An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading. Proc. Phys. Soc. London, 1949. B62: p. 676. [6] Ferreira, F., M.A. Vaz, and J.A. Simões, Mechanical properties of bovine cortical bone at high strain rate. Materials Characterization, August 2006. 57( 2): p. 71-79. [7] Harding, J., E.O. Wood, and J.D. Campbell, Tensile Testing of Materials at Impact Rates of Strain. Journal of Mechanical Engineering Science, 1960. 2: p. 88-96. [8] Duffy, J., J.D. Campbell, and R.H. Hawley, On the Use of a Torsional Split Hopkinson Bar to Study Rate Effects in 1100-0 Aluminum. J. Appl. Mech., 1971. 38(83-91). [9] Marquardt, D., An algorithm for least squares estimation of non-linear parameters J. Ind. Appl. Math. , 1963. 11(2): p. 431-441. [10].Levenberg, K., A Method for the Solution of Certain Non-Linear Problems in Least Squares. The Quarterly of Applied Mathematics, 1944. 2: p. 164-168.
ScienceDirect Procedia Engineering 114 (2015) 784 – 791
1st International Conference on Structural Integrity
Inverse Characterization of Material Constitutive Parameters for Dynamic Applications Inês Oliveiraa, Pedro Teixeiraa*, Fernando Ferreirab, Ana Reisa,c a
Institute of Science And Innovation in Mechanical and Industrial Engineering (INEGI), Rua Dr. Roberto Frias 400, 4200-465 Porto, Portugal b Polytechnic Institute of Porto (ISEP), Rua Dr. António Bernardino de Almeida 431, 4249-015 Porto, Portugal c Faculty of Engineering of the University of Porto (FEUP), Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Abstract Accurate modelling of mechanical dynamic behavior is a prerequisite for an effective analysis of both manufacturing processes and dynamic load during service life of the components. Consequently, precise and reliable rate dependent constitutive models, and related parameters, are required to represent the plastic behavior of metals undergoing high-strain rate deformations. However, these parameters are not always directly measurable. This study presents an inverse methodology to identify the dynamic properties of metallic materials by using the Split Hopkinson pressure bar. First, a symmetric finite element model for the simulation of SHPB test was implemented into ABAQUS/Explicit software. An inverse analysis methodology was then established, enabling the determination of the material parameters for the Johnson-Cook constitutive law. A user defined objective function was used to measure the optimality of the parameters, by evaluating the error between numerical and experimental results of SHPB test, namely the elastic strain wave’s evolution with time. The minimization of the objective function with respect to the material parameters was accomplished using the Levenberg–Marquardt optimization algorithm. ©2015 2015Published The Authors. Published by isElsevier © by Elsevier Ltd. This an open Ltd. access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering. Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering
Keywords: Hopkinson bar; High strain rate; Aluminum AA5182; Constitutive modeling; Finite element optimization
* Corresponding author. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: [email protected]
1877-7058 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering
doi:10.1016/j.proeng.2015.08.027
785
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
1. Introduction Computational simulation plays an indispensable role in the integrity analysis of components and structures during their manufacture and service life. Robust numerical models are needed to shorten design cycles, reduce manufacturing costs and create reliable final products. The accuracy of a FE simulation largely depends on the reliability of the input data, namely geometry, applied load, contact properties, boundary conditions and material data. Specifically, the knowledge of the material properties is of utmost importance to perform useful simulations. A correct material model requires not only an appropriate material constitutive law but also the determination of its parameters at different conditions. In line with these trends and needs, dynamic behavior of metallic materials has become one of the most current concerns. Besides, in the most of the applications of the metallic materials, e.g. industries as aerospace, military, civil engineering and automotive, the loading during service life has dynamic characteristics. There are several methods to determine dynamic material properties [1, 2]. The split Hopkinson pressure bar (SHPB) technique has become one of the most popular experimental tools for the study of dynamic behavior of solid materials. Attempts to apply the SHPB to the determination of constitutive laws parameters are presented by several authors [1, 2]. Typically, these parameters are estimated by the curve fitting method, using the stress-strain diagram derived from the recorded strain gage signals in a SHPB experiment. The engineering stress ( V s ) and strain ( H s ) as function of transmitted pulse strain are expressed by:
Vs
E
A Ht As
(1)
Hs
2C0 t H r dt L ³0
(2)
Hs
2C0 Hr L
(3)
where, E is the modulus of elasticity of the bar material, A is the cross-sectional area of the bar, As is the crosssectional area of the specimen, L is the gauge length of the specimen and C 0 is the speed of the stress wave in the pressure bar. H r and H t are the reflected and transmitted strains in the incident and transmitted bar, respectively. As verified, this strategy is highly dependent on constants used for the stress-strain computation. To reduce the user inputs and associated errors, it is proposed, in the present work, an alternative inverse approach. It is based on the quantitative assessment of the combination of constitutive parameters that leads to a minimum error between the time dependent elastic waves, acquired experimentally, and the equivalent ones attained by process simulation. In short words, the main idea of this approach is to perform a set of experiments, simulate them, and fit the material parameters in order to numerically obtain the same results as in the experimental ones, using as comparison the reflected and transmitted strains [3]. A set of split Hopkinson pressure bar tensile tests were conducted. Specimens, made on aluminum AA5182, were subjected to different strain-rates as result of the variation of the impact pressure and the geometrical properties (as the gage length). In terms of numerical work, an optimization procedure that makes use of MATLAB Levenberg– Marquardt routine was developed. This procedure uses a script to generate the input file for ABAQUS in order to obtain finite element solution for SHPB process. After simulation another executable file is used to read desirable data from ABAQUS output database. The material constitutive law and its parameters where introduced in the numerical analysis by using a user-defined VUHARD ABAQUS Subroutine. These steps were repeated allowing the iterative updating of the material parameters in the Finite Element (FE) model. The philosophy was to minimize the difference between numerically predicted and experimentally measured strain elastic waves according to a predefined error criterion.
786
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
2. Materials and Methods 2.1. Experimental Setup The first use of a long thin bar to measure the pulse shape induced by an impact is attributed to Hopkinson (1914) [4]. Kolsky, in 1949 [5], developed an experimental setup with two long bars and a short specimen, as is presented in Fig. 1.
Fig. 1. Schematic representation of a split Hopkinson pressure-bar device.
This traditional SHPB or Kolsky apparatus, used for dynamic compression experiments, consists of a striker, incident, and transmission bar. A gas gun, whose pressure controls the velocity of the striker bar, is used to drive the striker bar towards the incident bar. The impact of the striker bar on the incident bar is responsible for the creation of an elastic stress wave, which propagates trough the incident bar towards the specimen, which is sandwiched between the incident and the transmission bar. When stress wave reaches the incident bar/specimen interface, part of the stress wave is transferred to the specimen, causing a quick deformation, while part of the stress wave is reflected back to incident bar, called reflected wave [6]. The technique has been extended to tension [7] and to torsion [8]. In this work, dynamic tensile tests were performed using a tensile split Hopkinson pressure bar, whose setup is presented in Fig. 2.
Fig. 2. SHPB experimental setup. Grips Detail.
The setup of the Hopkinson bar used in this study includes two steel bars of 16 mm diameter and a gas gun equipped with a pneumatic valve. The incident (input) bar is 5735 mm long and the strain gauge located at 2400 mm from the impact end of the bar. The transmission (output) bar has a length of 2845 mm and the stain gauge positioned at 330 mm away from the specimen. The length of the striker bar is 1900 mm. Electronic equipment allows the measurement of the impact velocity. Signal conditioning with wide band amplification is also provided by adequate equipment to measure the dynamic strain in both bars. A LeCroy 9450A scope is used to record the signal at 2 MHz sample rate. The recorded signals are acquired by a personal computer through a RS232 connection. For testing metallic materials, the dog-bone shaped specimens were connected by screws to the bars grips. This assembling strategy ensures a good transmission of the waves [6]. The tested geometries are presented in Fig. 3.
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
Fig. 3. SHPB tensile specimen’s geometries. Dimensions are in millimetres.
2.2. Simulation Tool The commercial FEA software ABAQUS was used to model the split Hopkinson pressure bar. As referred, the experimental setup is composed of a cylindrical input bar of 16 mm diameter and 5735 mm length, a flat dog-bone specimen placed between the input bar and the output bar of 16 mm diameter and 2845 mm length. For simplicity, and due to symmetry, only one quarter of the problem was modelled and appropriate boundary conditions were imposed. In the present FE model, the specimen is considered as a deformable part, meshed with C3D8R elements, which is an eight node linear brick element, with reduced integration. The thickness of the specimen is equal to 1 mm and two elements were created in this direction. The bars, also meshed using C3D8R elements, were considered deformable parts, working only in the elastic regime. A total 12202 elements were considered in the FE model. The impact simulation was performed by applying a prescribed pressure on the bar free end. This pressure was calculated based on the incident wave acquired during the experiments. Element sets were created in both input and output bar, allowing the measure of elastic displacements in the same points as in the experimental tests. The simulation process time was fixed to 270 μs in agreement with experiments. A dynamic explicit time integration scheme (ABAQUS/Explicit) was employed
Fig. 4. Symmetric finite element model of the specimen and the pressure bars.
787
788
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
2.3. Optimization Procedure An inverse iterative FE technique was used to solve the problem of identifying the parameters for the JohnsonCook constitutive law, presented below.
V eq
§
§
©
©
A BH ¨¨1 C ln ¨ HH n
0
·· ¸ ¸¸ ¹¹
(4)
The parameters were estimated by minimizing the objective function, which expresses the discrepancy between the experimentally measured and the numerically computed strains at certain points of the bars. The mathematical model for inverse analysis can be expressed as: Find: A, B, n, C, H 0 Minimize: I ( A, B, n, C, H 0 ) With:
I
§ H iexp H inum · ¨ ¸ ¦ s i 1© ¹ s
2
(5)
where ݏis the number of experimental data points, H iexp the experimentally measured data point and H inum the numerically computed strain as a function of the unknown material parameters. The optimization problem expressed above was solved by the Levenberg-Marquardt (LM) algorithm. It consists in an iterative technique that finds the minimum of a function which is expressed as the sum of squares of nonlinear functions [9, 10]. It has become a standard technique for nonlinear least-squares problems and can be thought as a combination of steepest descent and the Gauss-Newton (GN) method. The procedure consists in a sequence of linear least squares approximations to the nonlinear problem. This algorithm presents a quadratic convergence that makes it very fast. However, the results highly depend on the initial parameters. As in the most of the cases, an accurate prediction of an appropriate set of initial parameters is not easy, being unrealistic for many applications. On the other hand, gradient descent algorithm is less dependent on the initial guess, but it is a first-order optimization algorithm, meaning that its speed of convergence is generally low. Therefore, the LevenbergMarquardt algorithm is a hybrid optimization technique which makes use of the positive attributes of both. In short, when the current solution is far from the correct one, the algorithm makes use of the steepest descent method properties, guaranteeing the convergence. When the current solution is close to the optimal solution, it uses the Gauss-Newton technique. The optimization procedure, summarized in the following diagram, was scripted under MATLAB. The optimization progresses from a given vector of initial material parameters towards the final ones, by solving the direct problem, defined by the SHPB finite element model. A developed Python script generates the input for ABAQUS. The material constitutive law is introduced in the numerical simulation using a user-defined FORTRAN subroutine, called VUHARD. Another Python script is used to read and extract the strain history from the ABAQUS output file. The MATLAB program reads both simulation and experimental results, and calculates the objective function, as explained above. The optimization process, governed by Levenberg-Marquardt algorithm, will continue until the objective function reaches the minimum specified by the convergence criterion. For each iteration, Johnson-Cook constitutive law parameters are updated and written by MATLAB to serve as ABAQUS inputs for the next direct problem.
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
Fig. 5. Flow-chart of the inverse method for material parameter identification.
3. Results and Discussion 3.1. Optimization results For the optimization algorithm, an initial set of parameters was defined by using a simple preliminary approximation. This preliminary approximation follows the procedure of the traditional approach, i.e. the stressstrain curves calculated based on strain-time evolution were used to perform an initial estimative of material parameters. The calculation process was stopped after 100 iterations or after the stop criterion was attained. The results of the optimization process are presented in Fig. 6 and 7. The parameters for the Johnson-Cook constitutive law obtained by the iterative program are summarized in Table 1. The time-strain evolutions obtained by the optimization procedure show some, but non-dramatic disparity, when compared with the wave achieved from experimental results, with especially significance in the duration of the output bar wave. The usual SHPB assumptions and simplifications can be the cause for the difference. Table 1: Identified parameters for the Johnson-Cook constitutive law Parameter
Value
A B n C H0
77.57 557.23 0.53 0.00067 0.00010
789
790
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
In addition, and as is possible to confirm in the Fig. 6 and 7, one of the major issues of the SHPB is the acquisition of data containing substantial measurement noises or errors, typically associated to experimental uncontrollable factors. The noise has a negative influence on the optimization procedure. The error (difference between both curves) is calculated at every point, meaning that it is substantially affected by the curve’s quality. However, despite the verified difference, both curves present the same general appearance and maximum values.
Fig. 6. Experimental and optimized numerical results for the geometry 1.
Fig. 7. Experimental and optimized numerical results for the geometry 2.
4. Summary and Conclusions Mechanical material characterization represents a research challenge, especially in the dynamic domain, since the mechanical properties of some materials are influenced by the loading rate. The main objective of this research work is to contribute to the numerical modelling improvement in the prediction of workpiece behaviour during dynamic processes. For such purpose, high strain rate tensile tests have been performed on AA 5182 specimens, by means of
Inês Oliveira et al. / Procedia Engineering 114 (2015) 784 – 791
791
a SHPB apparatus. Afterwards, the same tests have been numerically reproduced with a finite element commercial code. A material characterization methodology to determine the material parameters of a given material constitutive model from the given high strain rate experiment was then established. The characterization approach is based on an inverse technique in which an inverse problem is formulated and solved as an optimization procedure. The input of the optimization procedure is the characteristic signal from the high strain rate experiment, in this case, the strain-time evolution. The output of the procedure is the optimum set of material parameters determined by fitting a numerical simulation to the high strain rate experimental signal. This characterization procedure demonstrates advantages in comparison with other techniques. Numerical results of the characterization process showed that the implemented algorithm is effective in determining the Johnson-Cook material constants for the aluminium alloy 5182. At this stage, some words should be dedicated to possible strategies that can be used to improve the achieved results. In first place, more experimental tests should be conducted in order to validate the numerical model results and optimization process conclusions. More geometries and strain-rates should be investigated. Additionally, the approach must be extended to different materials laws and materials. An improvement of the presented optimization procedure is required, possibly by introducing a new method to perform an initial assessment of parameters, allowing a faster convergence of the results. Acknowledgements Authors gratefully acknowledge funding of Project SAESCTN-PII&DT/1/2011 co-financed by Programa Operacional Regional do Norte (ON.2 – O Novo Norte), under Quadro de Referência Estratégico Nacional (QREN), through Fundo Europeu de Desenvolvimento Regional (FEDER). Besides, the authors also gratefully acknowledge funding of Project EXPL/EMS-TEC/2419/2013, which has the financial support of FCT, Portuguese national funding agency for science, research and technology. References [1] Sasso, M., G. Newaz, and D. Amodio, Material characterization at high strain rate by Hopkinson bar tests and finite element optimization. Materials Science and Engineering: A, 25 July 2008. 487(1-2): p. 289-300. [2] El-Magd, E. and M. Abouridouane, Characterization, modelling and simulation of deformation and fracture behaviour of the light-weight wrought alloys under high strain rate loading. International Journal of Impact Engineering, May 2006. 32(5): p. 741–758. [3] Lee, O.S. and M.S. Kim, Dynamic material property characterization by using split Hopkinson pressure bar (SHPB) technique. Nuclear Engineering and Design, December 2003. 226(2): p. 119–125. [4] Hopkinson, B., A Method of Measuring the Pressure Produced in the Detonation of High Explosives or by the Impact of Bullets. Philos. Trans. R. Soc. (London) 1914. 213: p. 437-456. [5] Kolsky, H., An Investigation of the Mechanical Properties of Materials at Very High Rates of Loading. Proc. Phys. Soc. London, 1949. B62: p. 676. [6] Ferreira, F., M.A. Vaz, and J.A. Simões, Mechanical properties of bovine cortical bone at high strain rate. Materials Characterization, August 2006. 57( 2): p. 71-79. [7] Harding, J., E.O. Wood, and J.D. Campbell, Tensile Testing of Materials at Impact Rates of Strain. Journal of Mechanical Engineering Science, 1960. 2: p. 88-96. [8] Duffy, J., J.D. Campbell, and R.H. Hawley, On the Use of a Torsional Split Hopkinson Bar to Study Rate Effects in 1100-0 Aluminum. J. Appl. Mech., 1971. 38(83-91). [9] Marquardt, D., An algorithm for least squares estimation of non-linear parameters J. Ind. Appl. Math. , 1963. 11(2): p. 431-441. [10].Levenberg, K., A Method for the Solution of Certain Non-Linear Problems in Least Squares. The Quarterly of Applied Mathematics, 1944. 2: p. 164-168.