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A computational determination of the Cowper–Symonds parameters from a single Taylor test
Applied Mathematical Modelling 37 (2013) 4698–4708
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A computational determination of the Cowper–Symonds parameters from a single Taylor test C. Hernandez a, A. Maranon a,⇑, I.A. Ashcroft b, J.P. Casas-Rodriguez a a b
Mechanical Engineering Department, Universidad de los Andes, Cr 1 Este 18 A 70, Bogota, Colombia Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK
a r t i c l e
i n f o
Article history: Received 30 April 2012 Received in revised form 12 September 2012 Accepted 1 October 2012 Available online 11 October 2012 Keywords: Inverse problem Material characterization Taylor impact testing Genetic algorithms Multi-objective optimisation
a b s t r a c t A novel technique for the dynamic characterization of metals from a single Taylor impact test is proposed. This computational characterization procedure is based on the formulation and solution of a first class inverse problem, in which the silhouette of the Taylor specimen’s final shape is expressed as a vector of its geometrical moments and used as input parameter. The inverse characterization problem is reduced to an optimization problem where the optimum material parameters for the Cowper–Symonds material model are determined. The optimization process is performed by a range adaptation real-coded genetic algorithm. Numerical example for the characterisation of 1018 steel is implemented and presented to validate the methodology presented in this paper. The effectiveness and simplicity of the proposed characterization procedure makes it an appropriate tool for the characterization of metals at high strain rates. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Although the development of analytical and numerical approaches to understand problems involving high strain rates is still incomplete, they can provide a considerable insight during both the design, and the prediction of in-service performance of structures that absorb large amount of energy over short periods of time, e.g. vehicle and aircraft structures at impact. However, numerical approaches require the constitutive description of the nonlinear dynamic mechanical behavior of materials – usually in the form of constant parameters – in an appropriate analytical material model like the Johnson–Cook plasticity model or the Cowper–Symonds material model, among others. Depending on the strain rate to be applied to a particular material, there exist a number of experimental techniques that can be used to characterize the constant parameters in analytical material models, for example: Quasi-static tension test (104 to 100 s1, Split Hopkinson Pressure Bar (102 to 104 s1), Taylor Test (104 to 106 s1), and Inverse Flyer Plate 106 to 109 s1). In particular, the Taylor impact test [1] is a simple and inexpensive experiment where a flat-ended cylinder of a known initial uniform cross-sectional area is fired onto a rigid target. The terminal geometry of the deformed cylinder can be used to characterize computationally the material behavior at high strain rates. The computational characterization of materials, in terms of determining the material parameters, has been the subject of a number of research programs. In the first stage of the development of computational characterization methods, several researchers implemented quasi-static experimental test in combination with numerical simulation and optimization
⇑ Corresponding author. Tel.: +57 1 3324322; fax: +57 1 3324323. E-mail address: [email protected] (A. Maranon). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.10.010
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techniques to estimate the material parameters. Researchers such as Holmquist and Johnson [2], Mahnken and Stein [3,4], Meuwissen et al. [5], Kucharski and Mróz [6], Springmann and Kuna [7], among others, developed numericalexperimental procedures to determine the material parameters for diverse material models. These procedures usually used force–displacement curves as input data to determine the material parameters. In recent years, in response to the need for more advanced dynamic simulations, researchers turned their attention to dynamic material characterization. The proposed methodologies in this stage used experimental techniques that evidenced the strain rate dependency behavior of some materials and allowed the precise estimation of material constants. Soares et al. [8] presented a characterization technique to predict the mechanical properties of composite plates. The technique used an optimization process to minimize the difference between the numerically predicted natural frequencies with the measured response of vibrating plates. This technique was also used by Araújo et al. [9] to determine the elastic material modulus of composite plates. Such approaches were efficient but limited to elastic properties and were not able to estimate the strength material properties. Other techniques also used dynamic cyclic deformation to characterize materials. Yoshida et al. [10] proposed a method that used bending moments of metal sheets to model the strain hardening effect and to indentify the material parameters for a elasto–plastic material model. Shi et al. [11] implemented an inverse method to determine material parameters. The proposed method used a finite element model and measured resonance frequencies as a non destructive technique to characterize materials. In order to characterize materials at higher strain rates, experimental techniques such as Split Hopkinson Pressure Bar and Taylor test were implemented in computational characterization strategies. Sasso et al. [12] used experimental Split Hopkinson Pressure Bar tests to estimate material constants for steel. The procedure used a finite element model of the event and an optimization module to adjust the material constants. Parameters determined by this technique permitted to take into account the effects of strain and strain rate hardening and thermal softening. Milani et al. [13] proposed a methodology for determining the material parameters for the Johnson–Cook constitutive model. The strategy of characterization used the data from Split Hopkinson Pressure Bar tests at different strain rates and temperatures to simultaneously estimate model parameters. Allen et al. [14] and Rule [15] presented two similar approaches to determine the materials parameters from Taylor impact tests experiments. They developed numerical methodologies to obtain and refine constants for dynamic material strength models. The methodologies refined the material constants by optimizing the difference between the results of a series of Taylor test at different impact velocities and simulations of the event. The strategy minimized the volume difference to determine a more accurate set of material constants. The procedure was illustrated by obtaining material constants for a Johnson–Cook material model of copper. The optimization techniques used for material characterization procedures are diverse. Among them, the evolutionary computational techniques and machine learning techniques have proved to be reliable. Huber and Tsakmakis [16] employed neural networks to determine material parameters. This neural network used a number of finite element simulations of spherical indentations to adjust the material constants to match the response of materials to experimental data. Liu et al. [17] proposed a computational technique for material characterization of composites. A genetic algorithm in combination with the least squares method was used to optimize the dynamic displacement response of a composite plate. Han et al. [18] developed a procedure using neural networks to characterize properties of materials. The procedure consisted of material parameter recognition by using the response of composite cylinders when elastic waves were applied. Feng et al. [19] proposed a hybrid algorithm for the recognition of parameters for a visco–elastic material model. The hybrid algorithm used a combination of genetic algorithms and particle swarm optimization routines to determine the fittest set of material constants. Zain-ul-abdein et al. [20] used genetic algorithms to estimate the material constants of aluminum. The algorithm used as input data tensile test at different temperatures and strain rates. From the literature review, it is possible to establish that there are many numerical procedures and strategies proposed by a number of authors used for the estimation of material parameters. Those characterization procedures use diverse experimental techniques in combination with optimization strategies and, in some cases, numerical simulations to estimate the material parameters. In general, the parameters estimated by these procedures demonstrate good agreement when used in simulations and compared with experimental results. However, these techniques often show one or both of the following limitations: (1) several experimental tests are required in the characterization process, in many cases, complicated and expensive. (2) Material parameters are determined sequentially, dismissing interactions between parameters. This paper presents a novel technique for the dynamic characterization of metals based on the formulation and solution of a first class inverse problem [21]. The characterization procedure consists of the determination of the Cowper–Symonds material model constants from a single Taylor impact test. The input parameter of the procedure is the final shape of a Taylor impact test specimen, in terms of its silhouette’s central geometric moments, at a given impact velocity. The output parameters are the material model constants, which are determined by fitting the final shape of a numerically simulated Taylor specimen to the final shape of the experimental specimen. This optimization procedure is performed by a real-coded genetic algorithm. This paper is divided in six sections. First some preliminary concepts used during the development of this work are presented. Then, formulation of the characterization problem as an inverse problem is stated. In the fourth section, the methodology for the material characterization is proposed, followed by a numerical analysis of the performance of the procedure. Finally, results and conclusions are presented.
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2. Preliminary concepts 2.1. Taylor impact test The Taylor test, or cylinder impact test, is a very valuable experimental technique devised to estimate the average dynamic yield strength of materials at high strain rates, achieving strain rates in the order of 104 s1 to 106 s1. The test method was proposed by Taylor in 1948. It consists of impacting a right circular cylinder, at a known velocity, against a rigid barrier. The undeformed length, the final length and the deformed diameter are measured to compare the rod geometry before impact and after the rod has become to rest. With a simple analytical analysis, developed by Taylor [1] and then improved by Whiffin [22], is possible to estimate the dynamic yield strength. Some years later Hawkyard [23] enhanced the analysis assuming energy conservation and obtained better agreement at higher impact velocities. Although the state of stress and strain rate are not uniform along the specimen, a state of uniaxial stress and average strain rates are assumed to approximate analytically the properties of the material [24], and the plastically deformed region is assumed to be proportional to the yield stress of the material at the given strain rate. Although the analytical analysis of the Taylor test gained importance in the past, at the present the numerical analysis of the test has become a valuable tool to validate constitutive strength models by correlating the final cylinder length of experimental and numerical test. This approach was implemented by Johnson and Cook [25,26] and Johnson and Holmquist [27] to validate their proposed material model. On the other hand, Wilkins and Guinan [28] also simulated the Taylor test using a elasto–plastic model capable of showing the strain rate dependency of the dynamic yield strength. Simulations indicated that the plastic wave responsible for the mushrooming of the cylinder is not constant along the specimen. Some aspects in the numerical simulation of Taylor test are presented by Brüning and Driemeier [29]. They suggested that the simulation of the Taylor test requires a rate-dependent material model in order to accurately represent the strains occurring on the specimen. The simulations made by Brünig and Driemeier also provide evidences on the high dependence of the specimen’s final shape with the material parameters. 2.2. Cowper–Symonds material model The Cowper–Symonds material model [30] is a simple elasto–plastic, strain hardening, strain rate hardening model that uses the empirical formulation described by Ludwik [31], in which materials strengthen when plastic deformations are applied. This behavior is known as work hardening or strain hardening. The Cowper–Symonds material model scales the initial yield stress (ry ) by two factors. A strain factor and a strain rate factor as shown in Eq. (1) [30]. Where r0 is the initial yield stress, e_ is the strain rate, C and P are the Cowper–Symonds strain rate parameters, b is the strain hardening parameter, which adjust the contribution of isotropic and kinematic hardening, eeff p is the effective plastic strain, and Ep is the plastic hardening modulus which is given in terms of the elastic modulus E and the tangent elastic modulus Etan as shown in Eq. (2).
"
ry
# 1P e_ eff ¼ 1þ r0 þ bEp ep ; C
Ep ¼
Etan E : E Etan
ð1Þ
ð2Þ
The Cowper–Symonds model is a phenomenological material equation, thus the parameters have to be determined from experimental observations. In consequence, the performance of the material model rests on experimental data from which the parameters have been fitted. The ability to accurately describe the material behavior is a combined responsibility between the strength model selection and the values of the associated constants [2]. As show on the two previous equations, the implementation of this material model is linked with the determination of seven parameters: initial yield stress r0 , Cowper–Symonds strain rate parameters C and P, strain hardening parameter b, elastic modulus E, tangent elastic modulus Etan and the Poisson’s ratio l. The Cowper–Symonds model was adopted in this work because it is one of the typical material models used to represent the mechanical behavior of several materials at high strain rates. It shows the usual configuration of strain rate dependant phenomenological models in which the yield stress is modified by constant parameters. Then, the Cowper–Symonds model is a good example to show the performance of the proposed dynamic material characterization and the ability to determine the material parameters of a given material model. 2.3. Genetic algorithms Genetic Algorithms (GAs) are optimization and search techniques developed by Holland [32] based on the mechanics of natural selection and genetics [33]. Its concept is derived from the Darwin’s theory of survival of the fittest in combination with a structured stochastic process [34]. The GAs differ from the traditional derivative optimization techniques because of the use of a group of candidate solutions (population) that evolves in every iteration (generation) by combining (reproducing) the fittest individuals. The fitness of an individual is determined by computing a response function score. This fitness
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function is responsible for directing the search, as an alternative of derivation used in traditional optimization methods. These characteristics make GAs efficient optimization techniques when the analytical function of the problem is unknown or to explore complex (several local minima) search spaces. The mechanics of an essential GA consist, at least, of four basic steps. See Fig. 1. First, the initial population of candidate solutions is created randomly. In the second step, the fitness of each individual is evaluated. In the third step, the concept natural selection is applied. The individuals are selected according to their fitness value. Higher the fitness, higher the probability of being selected. In the fourth step, two operators are applied to generate the new population: crossover and mutation. The crossover operator recombines two of the selected individuals to produce two offspring that replace the parents and the mutation operator is used to increase the diversity in the new population by random modification of a particular individual. Steps two through four are repeated until a determined number of generations are completed or a convergence criterion is satisfied. 2.4. Geometric moments Geometric moments (GM) are invariants used in image analysis to represent patterns contained in images. This approach, proposed by Hu [35], consists of representing images by a set of its two-dimensional moments with respect to a fixed coordinate system. Since then, GM have been used in patter recognition, ship and aircraft identification and many image analysis applications. Given that an image can be represented as a density function f ðx; yÞ, mathematically GM of order ðp þ qÞ, are defined in terms of the Riemann integral as shown in Eq. (3).
mpq ¼
Z
1
Z
1
1
xp yq f ðx; yÞdx dy;
ð3Þ
1
where p; q ¼ 0; 1; 2; 3 . . . 1. A uniqueness and existence theorem, demonstrated by Hu [35], states that assuming that the density function f ðx; yÞ is a piecewise continuous bounded function and can have non zero values only in finite part of the xy plane; then moments mpq of all orders exists and are uniquely determined by f ðx; yÞ and conversely. To make the GM independent from the position of the image reference system, it is convenient to evaluate the moments with respect to the image centroid. These moments are called Central Geometric Moments (CGM) and are defined mathematically as shown in Eq. (4).
lpq ¼
Z
1
1
Z
1
Þq f ðx; yÞdx dy; ðx xÞp ðy y
ð4Þ
m01 : m00
ð5Þ
1
where
x ¼
m10 ; m00
¼ y
The CGM can be normalized [36] into the Normalized Central Geometric Moments (NGM) to make them invariant under translation and scale as shown in Eq. (6).
gpq ¼
l
l
pq ðpþqþ2Þ=2 00
ð6Þ
:
3. Formulation of material characterization as an inverse problem The material characterization problem, in terms of determining the material constants, can be formulated as an inverse problem and reformulated as an optimization problem. 3.1. Inverse problem input The input for the inverse problem is defined in terms of the final shape of a Taylor cylinder. Consider the cylinder, with specific initial diameter and length (D0 ; L0 ), after deforming by impacting against a rigid target at a known initial velocity (v 0 ). See Fig. 2. The cylinder material has an unknown set of constants. Let u~0 ðD0 ; L0 v 0 Þ stand for vector of the measured geometrical coordinates ðx; yÞ of the silhouette of the deformed specimen.
Initialization
Evaluation
Selection
New Population
Random initial population
Fitness for each individual
Parents for next generation
Crossover & Mutation
Fig. 1. Genetic algorithm flowchart.
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L0
D0
v0
Fig. 2. FE model of the Taylor impact test, (left) before and (rigth) after the impact.
3.2. Objective function Now, consider a finite element (FE) model representation of the Taylor event with the same initial geometry (D0 ; L0 ) and impacting at the same velocity (v 0 ). This FE model uses a known trial set of material constants (~ z) that characterize the material. Let u~0 ð~ z; D0 ; L0 v 0 Þ stand for the vector of geometrical coordinates of the computed silhouette of the deformed cylinder. Where the material constants set is: T ~ z ¼ ½r0 ; C; P; b; E; Etan ; l :
ð7Þ
Then, the inverse problem of material characterization can be formulated as follows: Given ~ uð~ z; D0 ; L0 v 0 Þ the coordinates of the silhouette of a cylinder after a Taylor impact test, find ~ z ¼ ½r0 ; C; P; b; E; Etan ; lT the optimal set of material parameters for the kinematic hardening material model. The optimal set of material constants is found when the objective error function, stated as a least-squares minimization problem, proposed in Eq. (8), is minimum.
zÞ ¼ u~0 ð~ z; D0 ; L0 v 0 Þ ~ /err ð~ uð~ z; D0 ; L0 v 0 Þ;
ð8Þ
where k k is the Euclidian norm of the vector. However, this formulation needs to be restated because of the great number of individual measurements of coordinates available for each silhouette and the difficulty to compare them. Then, the optimization problem can be restated in terms of the NGM of the cylinder silhouette coordinates as proposed in Eq. (9).
zÞ ¼ kg~0 ð~ /err ð~ uÞ gð~ uÞk;
ð9Þ
uÞ and gð~ uÞ are the vectors of the CGM of the computed and measured coordinates of the Taylor cylinder silhouwhere g~0 ð~ ettes respectively. This error function is often complicated and contains several local minima, therefore is not well suited to traditional gradient-based optimization techniques. For this reason, a derivative-free optimization process is more appropriate to implement in this particular application. From the wide field of derivative-free optimization techniques the genetic algorithms were selected due to their main features: Genetic algorithms are designed to exploit large highly complicated search spaces efficiently and yet, its implementation is very simple and computer efficient ([33]). The genetic algorithm formulation is explained in next section.
4. Methodology This paper presents a computational methodology for the dynamic characterization of metals from a single Taylor impact test. As shown in Fig. 3, the methodology consists of four basic steps: (1) a single Taylor test at known impact velocity is performed; (2) geometrical moments, up to fourth order, of the specimen’s silhouette are computed; (3) A Taylor test finite element (FE) model with trial material parameters is implemented in order to compare the final shape of the specimen against the experimental test; (4) a genetic algorithm optimization procedure is used to minimize the difference between the geometrical moments computed from the FE model with trial parameters, and geometrical moments from the measured experimental Taylor specimen’s silhouette.
4.1. Finite element model of Taylor test The FE model of the Taylor impact test was implemented using the softwate ANSYS/LS-DYNA employing an explicit solution scheme. Given the axisymmetric nature of the problem, the problem was modeled using axisymetric explicit 2D structural solid elements with six degrees of freedom at each node (PLANE162). The material model used for the Taylor specimen was the Cowper–Symonds material model discussed previously. The impact surface was assumed as a rigid body and the contact frictionless. Fig. 2 shows the FE simulation of the Taylor specimen before and after the impact.
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C. Hernandez et al. / Applied Mathematical Modelling 37 (2013) 4698–4708 INPUT Taylor impact test L0
Measured silhouette Geometric Moments
D0
Genetic Algorithm optimization
v0
OUTPUT Material Parameters
Objective function
Computed silhouette Geometric Moments
GA
FE model Taylor impact test
Fig. 3. Methodology for material characterization from a single Taylor impact test.
4.2. Geometrical moments computation The silhouette of the deformed Taylor specimen can be represented by a function f ðxÞ in which the diameter on each point is a function of the length along the specimen as shown in Fig. 4. This function is obtained by measuring different radius along the specimen length and interpolating a spline function. This spline function is smooth, differentiable, bounded and defined in a finite interval which makes it suitable to compute its geometrical moments. Due to the discrete nature of the FE model and measurements of problem, the expressions on Eqs. (3) and (4) should be approximated by a summation as shown in Eqs. (10) and (11) respectively. Where Dx is the element size in the length direction and N is the number of elements along the specimen’s length. The normalized central geometric moments up to fourth order were computed along with the normalized central geometric moments of the derivative of shape function up to fourth order. The geometric moments of the silhouette derivative are included because they introduce additional information regarding the shape of the specimen and facilitate the comparison between shapes improving the optimization process. The silhouette of the deformed Taylor specimen can be represented by a function in which the diameter on each point is a function of the length along the specimen as shown in Fig. 4, which facilitates the computation of its geometrical moments. Due to the discrete nature of the FE model and the 2D character of the problem, the expressions on Eqs. (3) and (4) should be approximated by a summation as shown in Eqs. (10) and (11), respectively. Where Dx is the element size in the length direction and N is the number of elements along the specimen’s length.
mpq ¼
N X xp f ðxÞqþ1 Dx;
ð10Þ
i¼1
lpq ¼
N X Þq f ðxÞDx: ðx xÞp ðf ðxÞ y i¼1
Fig. 4. Computation of geometrical moments of the deformed Taylor specimen’s silhouette.
ð11Þ
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4.3. Genetic algorithm optimization The optimization of the difference between the experimental and FE geometrical moments was performed by a GA optimization routine. In this optimization problem there are seven genes given by the material parameters that must be determined: ~ z ¼ ½r0 ; C; P; b; E; Etan ; lT . Traditionally, GAs use a binary codification to represent the genes to form a string and then apply the genetic operations. However, other coding types, such as the real codification have been considered to represent the genes. In particular, the real-coded GAs have shown that are more efficient than the binary-coded GAs when optimizing continuous and discrete variables [37]. For this reason real-coded representation of the genes was used is this optimization procedure. The real-coded adaptive range genetic algorithm implemented to characterize the material parameters is shown in Fig. 5 and its operators are described below. 4.3.1. Initialization The first generation of possible solutions is generated by dividing the search space of each parameter uniformly, and then the individuals are generated by the random combination of these parameters. 4.3.2. Evaluation The aptitude of each individual as a solution of the problems is determined by evaluating a fitness function. The fitness function w for every individual is given by Eq. (12), where /err is the objective function defined in Eq. (9).
wð/err Þ ¼
1 : zÞ 1 þ /err ð~
ð12Þ
4.3.3. Selection The selection operator implemented is the roulette wheel selection. This selection operator is based on a fitness proportionate method in which the probability of each individual to be selected as a parent for the next generation is associated to its fitness. Consequently, the highest the fitness of an individual, biggest the probability of being selected as a parent. 4.3.4. New population The new population is generated by applying two operators, crossover and mutation. Three crossover operators are used: heuristic, arithmetic and uniform. These operators are described by Adewuya [38] and the performance for real-coded genetic algorithms noted. The mutation operator used is uniform. Each of the individuals has the exactly equal chance of undergoing mutation. 4.3.5. Range adaptation On the other hand, a range adaptation operator is implemented to prevent the fast convergence to a local minima and loss of diversity. This operator, as shown by Maranon et al. [39], re-generate the entire population every M generations by calculating a new search space range and by generating a new population distributed accordingly to the new parameter range. The new range limits of each parameter is given by the average of the top half of individual in the previous generation, the standard deviation and a parameter k that determines the amplitude of the new range as shown in Eq. (13).
li ¼ xi kr1 ;
ð13Þ
where li are the new range limits for each parameters, xi and individuals of the previous generation for each parameter.
ri are the average and standard deviation of the top half of
5. Numerical analysis The performance and feasibility of the proposed characterization procedure is examined through numerical simulations in which the characterization of 1018 steel was performed. The numerical example consists of a finite elements simulation of a Taylor test specimen using the material constants shown in Table 1. The initial geometry of the specimen is 20 mm length
Every M generations
Range Adaptation New population
Initialization
Evaluation
Selection
New Population
Random initial population
Fitness for each individual
Parents for next generation
Crossover & Mutation
Fig. 5. Real-coded adaptive range genetic algorithm flowchart.
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C. Hernandez et al. / Applied Mathematical Modelling 37 (2013) 4698–4708 Table 1 Material parameters for 1018 steel [40].
l
E 200 GPa
r0
0.27
Etan
310 MPa
C
b
763 MPa
0.3
40 s
1
P
ef
5.0
0.75
Fig. 6. Results of the characterization process for run 4. (—) Taylor specimen shape (- -) predicted specimen shape.
Table 2 Results of characterization process (Five consecutive runs). E [GPa] 200
l [–]
r0 [MPa]
Run
0.27
310
2 3 4 5
195.8 195.5 200.3 199.0 195.7
0.277 0.273 0.280 0.269 0.270
Avg. error
1.4%
(2.1%) (2.2%) (0.2%) (0.5%) (2.1%)
1.6%
(2.6%) (1.1%) (3.7%) (0.4%) (0.0%)
314.4 298.0 289.2 329.0 337.5 5.4%
(1.4%) (3.8%) (6.7%) (6.1%) (8.9%)
Etan [MPa] 763
b [–] 0.30
728.4 702.1 703.0 769.0 857.0
0.340 0.305 0.259 0.337 0.317
6.7%
(4.5%) (8.0%) (7.9%) (0.8%) (12.3%)
9.3%
(13.3%) (1.7%) (13.7%) (12.3%) (5.7%)
C [s1] 40
P [–] 5.0
42.2 43.0 41.2 37.3 42.0
5.05 4.74 4.59 5.12 5.44
5.6%
(5.7%) (7.1%) (3.1%) (6.6%) (5.0%)
Fitness (1.0%) (5.2%) (8.2%) (2.4%) (8.8%)
0.996 0.993 0.995 0.996 0.996
5.12%
and 8 mm diameter. The impact velocity is 250 m/s. After impact the specimen length is 17.6 mm and the undeformed section length is 8.1 mm. The final shape of the Taylor specimen is shown in Fig. 6. The silhouette of the simulated specimen’s deformed shape is used as input parameter in the optimization process. Therefore, the optimization procedure target parameters are shown in Table 1. Normalized central geometrical moments, of the deformed specimen shape, up to fourth order were calculated using Eqs. (4) and (11); and in order to increase the reliance of the optimization procedure on the specimen’s shape, normalized central geometrical moments of the diameter function derivative were also computed. The 30 normalized central moments (15 from the diameter function and 15 from the derivative) were stored in a vector ~ g and used during the optimization in the fitness function. Five runs were performed to determine the efficiency of the characterization procedure. The GA was configured to use 60 individuals and 100 generations. The selection operator used in this work is the roulette wheel combined with elitism strategy. The new individuals are created by using one of three crossover operators implemented: heuristic, arithmetic or uniform. The crossover operator used to create each of the new offspring is selected randomly. The individuals are mutated using a uniform operator with mutation probability of 80%. The range adaptation operator was set to re-generate the entire population every 5 generations with amplitude parameter k of 1.5 standard deviations.
6. Results The feasibility and the effectiveness of the proposed characterization procedure were investigated via numerical simulations. The objective of the characterization example was to determine the Cowper–Symonds material parameters ~ z for the 1018 steel. The procedure used as input the silhouette of a Taylor specimen’s FE model. Five characterization replications were performed and its results are along with the relative estimation error are shown in Table 2. The material parameters were predicted with accuracy above 90% in the five examples. Seven of the eigth parameters were determined with relative error around 5%, while parameter b could only be determined with relative error around 10%. Typical graphical results obtained during this study are shown in Fig. 6. The graphic shows the comparison between the shape of the Taylor specimen
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Fig. 7. Convergence history of parameters for run 4.
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(—) and the shape of the specimen using the determined parameters (- -). In general, the final shape of the specimen is captured by the proposed characterization procedure, however, small differences between the Taylor specimen silhouettes are observed. These small differences cannot be captured by the geometrical moments used to represent the silhouette. To capture these small differences the order of the geometrical moments used must be increased, expanding the risk of amplifying the noise without the certainty of improving the characterization results. Fig. 7 show the typical convergence history of each of the material parameters as the genetic algorithm evolves, in particular the graphs show the results for run 4 and the evolution of the fitness. The charts show the evolution of each estimated material constant (—), the reference original material constant (-.-), and the search space limits (- -). The main finding of the numerical analysis is that the proposed characterization methodology can determine accurately and consistently the material parameters from a single Taylor impact test. The accuracy was observed in the fact that average global error was maintained below 10%, and the consistency was observed when during consecutive runs the results were kept uniform. Compared with other characterization methods explored in the literature, this proposed method showed three main advantages. The first advantage lies in the simplicity of the method. The use of a single Taylor impact test makes the parameters determination process inexpensive and relatively easy to apply. The second advantage is the simultaneous estimation of the material parameters. Traditional parameter estimation methods uncouple the material model to determine each of the material constants independently ignoring the synergy among parameters. The third advantage is that the characterization process is performed at high strain rates. The materials constants can be determined with impact experiments that undergo strain rates in a broad range from 104 s1 to 106 s1. However, due to the fact that the material parameters are determined using one test at a single strain rate; it is important to identify the strain rate of the particular application in which the material parameters are being use, and perform the experimental Taylor test at this particular average strain rate to obtain the best performance of the material model.
7. Conclusions In this paper a dynamic characterization computational procedure to determine the material parameters from a single Taylor impact test was proposed. The procedure involves the formulation and solution of an inverse problem to determine the Cowper–Symonds parameter of metals using as input the silhouette of a deformed Taylor specimen. Geometrical moments are used to represent the shape of the specimen and a genetic algorithm optimization procedure is used to optimize the set of parameter constants. The proposed characterization technique shows advantages in the material parameters estimation process compared with other characterization strategies: (1) Simplicity and easiness due to the use of a single Taylor test as experimental data. (2) Simultaneous estimation of material parameters that allows observing the interaction among parameters, unlike traditional methods that determine parameters sequentially. (3) Material parameters are determined at high strain rates, in the order of 104 s1 to 106 s1, making this procedure ideal for the development of accurate material models used in numerical simulations of dynamic and impact events. Numerical characterization examples were performed to test the effectiveness of the proposed characterization method. Numerical examples showed that the procedure is capable of capturing the shape of the Taylor specimen and determine the material parameters with accuracy above 90%. Results showed that the algorithm is effective and efficient. More studies need to be conducted to evaluate the performance of the procedure with other materials and experimental Taylor impact tests.
References [1] G. Taylor, The use of flat-ended projectiles for determining dynamic yield stress. i. theoretical considerations, Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 194 (1038) (1948) 289–299. [2] T. Holmquist, G. Johnson, Determination of constants and comparison of results for various constitutive models, J. Phys. IV France 1 (8) (1991) 853– 860. [3] R. Mahnken, E. Stein, A unified approach for parameter identification of inelastic material models in the frame of the finite element method, Comput. Methods Appl. Mech. Eng. 136 (3–4) (1996) 225–258. [4] R. Mahnken, E. Stein, Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations, Int. J. Plasticity 12 (4) (1996) 451–479. [5] M.H.H. Meuwissen, C.W.J. Oomens, F.P.T. Baaijens, R. Petterson, J.D. Janssen, Determination of the elasto–plastic properties of aluminium using a mixed numerical–experimental method, J. Mater. Process. Technol. 75 (1–3) (1998) 204–211. [6] S. Kucharski, Z. Mróz, Identification of plastic hardening parameters of metals from spherical indentation tests, Mater. Sci. Eng. A 318 (1–2) (2001) 65– 76. [7] M. Springmann, M. Kuna, Identification of material parameters of the Gurson–Tvergaard– eedleman model by combined experimental and numerical techniques, Comput. Mater. Sci. 32 (3–4) (2005) 544–552. [8] C. Soares, M. de Freitas, A. Araújo, P. Pedersen, Identification of material properties of composite plate specimens, Compos. Struct. 25 (1–4) (1993) 277– 285. [9] A.L. Araújo, C.M.M. Soares, M.J.M. de Freitas, P. Pedersen, J. Herskovits, Combined numerical–experimental model for the identification of mechanical properties of laminated structures, Compos. Struct. 50 (4) (2000) 363–372. [10] F. Yoshida, M. Urabe, V. Toropov, Identification of material parameters in constitutive model for sheet metals from cyclic bending tests, Int. J. Mech. Sci. 40 (2–3) (1998) 237–249. [11] Y. Shi, H. Sol, H. Hua, Material parameter identification of sandwich beams by an inverse method, J. Sound Vib. 290 (3–5) (2006) 1234–1255. [12] M. Sasso, G. Newaz, D. Amodio, Material characterization at high strain rate by Hopkinson bar tests and finite element optimization, Mater. Sci. Eng.: A 487 (1–2) (2008) 289–300.
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[13] A. Milani, W. Dabboussi, J. Nemes, R. Abeyaratne, An improved multi-objective identification of Johnson–Cook material parameters, Int. J. Impact Eng. 36 (2) (2009) 294–302. [14] D. Allen, W. Rule, S. Jones, Optimizing material strength constants numerically extracted from Taylor impact data, Exp. Mech. 37 (1997) 333–338. [15] W.K. Rule, A numerical scheme for extracting strength model coefficients from Taylor test data, Int. J. Impact Eng. 19 (9–10) (1997) 797–810. [16] N. Huber, C. Tsakmakis, Determination of constitutive properties from spherical indentation data using neural networks. Part i: the case of pure kinematic hardening in plasticity laws, J. Mech. Phys. Solids 47 (7) (1999) 1569–1588. [17] G.R. Liu, X. Han, K.Y. Lam, A combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves, Comput. Methods Appl. Mech. Eng. 191 (17–18) (2002) 1909–1921. [18] X. Han, D. Xu, G.R. Liu, A computational inverse technique for material characterization of a functionally graded cylinder using a progressive neural network, Neurocomputing 51 (2003) 341–360. [19] X.-T. Feng, B.-R. Chen, C. Yang, H. Zhou, X. Ding, Identification of visco–elastic models for rocks using genetic programming coupled with the modified particle swarm optimization algorithm, Int. J. Rock Mech. Min. Sci. 43 (5) (2006) 789–801. [20] M. Zain-ul-abdein, D. NTlias, J.-F. Jullien, A.I. Wagan, Thermo-mechanical characterisation of AA 6056–T4 and estimation of its material properties using genetic algorithm, Mater. Design 31 (9) (2010) 4302–4311. [21] N. Zabaras, K. Woodbry, M. Raynaud, Inverse Problems in Engineering: Theory and Practice, The American Society of Mechanical Engineers, New York, 1993. [22] A.C. Whiffin, The use of flat-ended projectiles for determining dynamic yield stress. II. Tests on various metallic materials, Proc. R. Soc. Lond. A 194 (1948) 300–322. [23] J. Hawkyard, A theory for the mushrooming of flat-ended projectiles impinging on a flat rigid anvil, using energy considerations, Int. J. Mech. Sci. 11 (3) (1969) 313–324. IN3–IN4, 325–333. [24] M. Meyers, Dynamic Behavior of Materials, Wiley Interscience, 1994. [25] G. Johnson, W. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceedings of the Seventh International Symposium on Ballistics, The hague, Netherlands, 1983, pp. 541–547. [26] G.R. Johnson, W.H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Fract. Mech. 21 (1) (1985) 31–48. [27] G.R. Johnson, T.J. Holmquist, Evaluation of cylinder-impact test data for constitutive model constants, J. Appl. Phys. 64 (8) (1988) 3901–3910. [28] M.L. Wilkins, M.W. Guinan, Impact of cylinders on a rigid boundary, J. Appl. Phys. 44 (3) (1973) 1200–1206. [29] M. Brnnig, L. Driemeier, Numerical simulation of Taylor impact tests, Int. J. Plasticity 23 (12) (2007) 1979–2003. [30] G. Cowper, P. Symonds, Strain hardening and strain-rate effects in the impact loading of cantilever beams, Tech. Rep., Brown University Division of Applied Mathematics, 1957. [31] P. Ludwik, Elemente der technologischen Mechanik, Springer, 1909. [32] J.H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, The MIT Press, 1992. [33] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, first ed., Addison-Wesley Professional, 1989. [34] R. Shaffer, G. Small, Learning optimization from nature: simulated annealing and genetic algorithms, Anal. Chem. 69 (1997) 236A–242A. [35] M.-K. Hu, Visual pattern recognition by moment invariants, IEEE Trans. Inform. Theory 8 (2) (1962) 179–187. [36] R. Mukundan, K. Ramakrishnan, Moment Functions in Image Analysis: Theory and Applications, World Scientific Pub. Co. Inc., 1998. [37] C. Janikow, Z. Michalewicz, An experimental comparison of binary and floating point representations in genetic algorithms, Proceedings of the Fourth International Conference on Genetic Algorithms, vol. 31, Morgan Kaufmann, 1991. [38] A.A. Adewuya, New methods in genetic search with real-valued chromosomes, Master’s thesis, Massachusetts Institute of Technology, 1996. [39] A. Maranon, A.D. Nurse, J.M. Huntley, Characterization of a single delamination using geometric moments and genetic algorithms, Opt. Eng. 42 (5) (2003) 1328–1336. [40] ANSYS, ANSYS LS-DYNA User’s Guide: ANSYS release 12.0 (2009).
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A computational determination of the Cowper–Symonds parameters from a single Taylor test C. Hernandez a, A. Maranon a,⇑, I.A. Ashcroft b, J.P. Casas-Rodriguez a a b
Mechanical Engineering Department, Universidad de los Andes, Cr 1 Este 18 A 70, Bogota, Colombia Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK
a r t i c l e
i n f o
Article history: Received 30 April 2012 Received in revised form 12 September 2012 Accepted 1 October 2012 Available online 11 October 2012 Keywords: Inverse problem Material characterization Taylor impact testing Genetic algorithms Multi-objective optimisation
a b s t r a c t A novel technique for the dynamic characterization of metals from a single Taylor impact test is proposed. This computational characterization procedure is based on the formulation and solution of a first class inverse problem, in which the silhouette of the Taylor specimen’s final shape is expressed as a vector of its geometrical moments and used as input parameter. The inverse characterization problem is reduced to an optimization problem where the optimum material parameters for the Cowper–Symonds material model are determined. The optimization process is performed by a range adaptation real-coded genetic algorithm. Numerical example for the characterisation of 1018 steel is implemented and presented to validate the methodology presented in this paper. The effectiveness and simplicity of the proposed characterization procedure makes it an appropriate tool for the characterization of metals at high strain rates. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Although the development of analytical and numerical approaches to understand problems involving high strain rates is still incomplete, they can provide a considerable insight during both the design, and the prediction of in-service performance of structures that absorb large amount of energy over short periods of time, e.g. vehicle and aircraft structures at impact. However, numerical approaches require the constitutive description of the nonlinear dynamic mechanical behavior of materials – usually in the form of constant parameters – in an appropriate analytical material model like the Johnson–Cook plasticity model or the Cowper–Symonds material model, among others. Depending on the strain rate to be applied to a particular material, there exist a number of experimental techniques that can be used to characterize the constant parameters in analytical material models, for example: Quasi-static tension test (104 to 100 s1, Split Hopkinson Pressure Bar (102 to 104 s1), Taylor Test (104 to 106 s1), and Inverse Flyer Plate 106 to 109 s1). In particular, the Taylor impact test [1] is a simple and inexpensive experiment where a flat-ended cylinder of a known initial uniform cross-sectional area is fired onto a rigid target. The terminal geometry of the deformed cylinder can be used to characterize computationally the material behavior at high strain rates. The computational characterization of materials, in terms of determining the material parameters, has been the subject of a number of research programs. In the first stage of the development of computational characterization methods, several researchers implemented quasi-static experimental test in combination with numerical simulation and optimization
⇑ Corresponding author. Tel.: +57 1 3324322; fax: +57 1 3324323. E-mail address: [email protected] (A. Maranon). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.10.010
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techniques to estimate the material parameters. Researchers such as Holmquist and Johnson [2], Mahnken and Stein [3,4], Meuwissen et al. [5], Kucharski and Mróz [6], Springmann and Kuna [7], among others, developed numericalexperimental procedures to determine the material parameters for diverse material models. These procedures usually used force–displacement curves as input data to determine the material parameters. In recent years, in response to the need for more advanced dynamic simulations, researchers turned their attention to dynamic material characterization. The proposed methodologies in this stage used experimental techniques that evidenced the strain rate dependency behavior of some materials and allowed the precise estimation of material constants. Soares et al. [8] presented a characterization technique to predict the mechanical properties of composite plates. The technique used an optimization process to minimize the difference between the numerically predicted natural frequencies with the measured response of vibrating plates. This technique was also used by Araújo et al. [9] to determine the elastic material modulus of composite plates. Such approaches were efficient but limited to elastic properties and were not able to estimate the strength material properties. Other techniques also used dynamic cyclic deformation to characterize materials. Yoshida et al. [10] proposed a method that used bending moments of metal sheets to model the strain hardening effect and to indentify the material parameters for a elasto–plastic material model. Shi et al. [11] implemented an inverse method to determine material parameters. The proposed method used a finite element model and measured resonance frequencies as a non destructive technique to characterize materials. In order to characterize materials at higher strain rates, experimental techniques such as Split Hopkinson Pressure Bar and Taylor test were implemented in computational characterization strategies. Sasso et al. [12] used experimental Split Hopkinson Pressure Bar tests to estimate material constants for steel. The procedure used a finite element model of the event and an optimization module to adjust the material constants. Parameters determined by this technique permitted to take into account the effects of strain and strain rate hardening and thermal softening. Milani et al. [13] proposed a methodology for determining the material parameters for the Johnson–Cook constitutive model. The strategy of characterization used the data from Split Hopkinson Pressure Bar tests at different strain rates and temperatures to simultaneously estimate model parameters. Allen et al. [14] and Rule [15] presented two similar approaches to determine the materials parameters from Taylor impact tests experiments. They developed numerical methodologies to obtain and refine constants for dynamic material strength models. The methodologies refined the material constants by optimizing the difference between the results of a series of Taylor test at different impact velocities and simulations of the event. The strategy minimized the volume difference to determine a more accurate set of material constants. The procedure was illustrated by obtaining material constants for a Johnson–Cook material model of copper. The optimization techniques used for material characterization procedures are diverse. Among them, the evolutionary computational techniques and machine learning techniques have proved to be reliable. Huber and Tsakmakis [16] employed neural networks to determine material parameters. This neural network used a number of finite element simulations of spherical indentations to adjust the material constants to match the response of materials to experimental data. Liu et al. [17] proposed a computational technique for material characterization of composites. A genetic algorithm in combination with the least squares method was used to optimize the dynamic displacement response of a composite plate. Han et al. [18] developed a procedure using neural networks to characterize properties of materials. The procedure consisted of material parameter recognition by using the response of composite cylinders when elastic waves were applied. Feng et al. [19] proposed a hybrid algorithm for the recognition of parameters for a visco–elastic material model. The hybrid algorithm used a combination of genetic algorithms and particle swarm optimization routines to determine the fittest set of material constants. Zain-ul-abdein et al. [20] used genetic algorithms to estimate the material constants of aluminum. The algorithm used as input data tensile test at different temperatures and strain rates. From the literature review, it is possible to establish that there are many numerical procedures and strategies proposed by a number of authors used for the estimation of material parameters. Those characterization procedures use diverse experimental techniques in combination with optimization strategies and, in some cases, numerical simulations to estimate the material parameters. In general, the parameters estimated by these procedures demonstrate good agreement when used in simulations and compared with experimental results. However, these techniques often show one or both of the following limitations: (1) several experimental tests are required in the characterization process, in many cases, complicated and expensive. (2) Material parameters are determined sequentially, dismissing interactions between parameters. This paper presents a novel technique for the dynamic characterization of metals based on the formulation and solution of a first class inverse problem [21]. The characterization procedure consists of the determination of the Cowper–Symonds material model constants from a single Taylor impact test. The input parameter of the procedure is the final shape of a Taylor impact test specimen, in terms of its silhouette’s central geometric moments, at a given impact velocity. The output parameters are the material model constants, which are determined by fitting the final shape of a numerically simulated Taylor specimen to the final shape of the experimental specimen. This optimization procedure is performed by a real-coded genetic algorithm. This paper is divided in six sections. First some preliminary concepts used during the development of this work are presented. Then, formulation of the characterization problem as an inverse problem is stated. In the fourth section, the methodology for the material characterization is proposed, followed by a numerical analysis of the performance of the procedure. Finally, results and conclusions are presented.
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2. Preliminary concepts 2.1. Taylor impact test The Taylor test, or cylinder impact test, is a very valuable experimental technique devised to estimate the average dynamic yield strength of materials at high strain rates, achieving strain rates in the order of 104 s1 to 106 s1. The test method was proposed by Taylor in 1948. It consists of impacting a right circular cylinder, at a known velocity, against a rigid barrier. The undeformed length, the final length and the deformed diameter are measured to compare the rod geometry before impact and after the rod has become to rest. With a simple analytical analysis, developed by Taylor [1] and then improved by Whiffin [22], is possible to estimate the dynamic yield strength. Some years later Hawkyard [23] enhanced the analysis assuming energy conservation and obtained better agreement at higher impact velocities. Although the state of stress and strain rate are not uniform along the specimen, a state of uniaxial stress and average strain rates are assumed to approximate analytically the properties of the material [24], and the plastically deformed region is assumed to be proportional to the yield stress of the material at the given strain rate. Although the analytical analysis of the Taylor test gained importance in the past, at the present the numerical analysis of the test has become a valuable tool to validate constitutive strength models by correlating the final cylinder length of experimental and numerical test. This approach was implemented by Johnson and Cook [25,26] and Johnson and Holmquist [27] to validate their proposed material model. On the other hand, Wilkins and Guinan [28] also simulated the Taylor test using a elasto–plastic model capable of showing the strain rate dependency of the dynamic yield strength. Simulations indicated that the plastic wave responsible for the mushrooming of the cylinder is not constant along the specimen. Some aspects in the numerical simulation of Taylor test are presented by Brüning and Driemeier [29]. They suggested that the simulation of the Taylor test requires a rate-dependent material model in order to accurately represent the strains occurring on the specimen. The simulations made by Brünig and Driemeier also provide evidences on the high dependence of the specimen’s final shape with the material parameters. 2.2. Cowper–Symonds material model The Cowper–Symonds material model [30] is a simple elasto–plastic, strain hardening, strain rate hardening model that uses the empirical formulation described by Ludwik [31], in which materials strengthen when plastic deformations are applied. This behavior is known as work hardening or strain hardening. The Cowper–Symonds material model scales the initial yield stress (ry ) by two factors. A strain factor and a strain rate factor as shown in Eq. (1) [30]. Where r0 is the initial yield stress, e_ is the strain rate, C and P are the Cowper–Symonds strain rate parameters, b is the strain hardening parameter, which adjust the contribution of isotropic and kinematic hardening, eeff p is the effective plastic strain, and Ep is the plastic hardening modulus which is given in terms of the elastic modulus E and the tangent elastic modulus Etan as shown in Eq. (2).
"
ry
# 1P e_ eff ¼ 1þ r0 þ bEp ep ; C
Ep ¼
Etan E : E Etan
ð1Þ
ð2Þ
The Cowper–Symonds model is a phenomenological material equation, thus the parameters have to be determined from experimental observations. In consequence, the performance of the material model rests on experimental data from which the parameters have been fitted. The ability to accurately describe the material behavior is a combined responsibility between the strength model selection and the values of the associated constants [2]. As show on the two previous equations, the implementation of this material model is linked with the determination of seven parameters: initial yield stress r0 , Cowper–Symonds strain rate parameters C and P, strain hardening parameter b, elastic modulus E, tangent elastic modulus Etan and the Poisson’s ratio l. The Cowper–Symonds model was adopted in this work because it is one of the typical material models used to represent the mechanical behavior of several materials at high strain rates. It shows the usual configuration of strain rate dependant phenomenological models in which the yield stress is modified by constant parameters. Then, the Cowper–Symonds model is a good example to show the performance of the proposed dynamic material characterization and the ability to determine the material parameters of a given material model. 2.3. Genetic algorithms Genetic Algorithms (GAs) are optimization and search techniques developed by Holland [32] based on the mechanics of natural selection and genetics [33]. Its concept is derived from the Darwin’s theory of survival of the fittest in combination with a structured stochastic process [34]. The GAs differ from the traditional derivative optimization techniques because of the use of a group of candidate solutions (population) that evolves in every iteration (generation) by combining (reproducing) the fittest individuals. The fitness of an individual is determined by computing a response function score. This fitness
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function is responsible for directing the search, as an alternative of derivation used in traditional optimization methods. These characteristics make GAs efficient optimization techniques when the analytical function of the problem is unknown or to explore complex (several local minima) search spaces. The mechanics of an essential GA consist, at least, of four basic steps. See Fig. 1. First, the initial population of candidate solutions is created randomly. In the second step, the fitness of each individual is evaluated. In the third step, the concept natural selection is applied. The individuals are selected according to their fitness value. Higher the fitness, higher the probability of being selected. In the fourth step, two operators are applied to generate the new population: crossover and mutation. The crossover operator recombines two of the selected individuals to produce two offspring that replace the parents and the mutation operator is used to increase the diversity in the new population by random modification of a particular individual. Steps two through four are repeated until a determined number of generations are completed or a convergence criterion is satisfied. 2.4. Geometric moments Geometric moments (GM) are invariants used in image analysis to represent patterns contained in images. This approach, proposed by Hu [35], consists of representing images by a set of its two-dimensional moments with respect to a fixed coordinate system. Since then, GM have been used in patter recognition, ship and aircraft identification and many image analysis applications. Given that an image can be represented as a density function f ðx; yÞ, mathematically GM of order ðp þ qÞ, are defined in terms of the Riemann integral as shown in Eq. (3).
mpq ¼
Z
1
Z
1
1
xp yq f ðx; yÞdx dy;
ð3Þ
1
where p; q ¼ 0; 1; 2; 3 . . . 1. A uniqueness and existence theorem, demonstrated by Hu [35], states that assuming that the density function f ðx; yÞ is a piecewise continuous bounded function and can have non zero values only in finite part of the xy plane; then moments mpq of all orders exists and are uniquely determined by f ðx; yÞ and conversely. To make the GM independent from the position of the image reference system, it is convenient to evaluate the moments with respect to the image centroid. These moments are called Central Geometric Moments (CGM) and are defined mathematically as shown in Eq. (4).
lpq ¼
Z
1
1
Z
1
Þq f ðx; yÞdx dy; ðx xÞp ðy y
ð4Þ
m01 : m00
ð5Þ
1
where
x ¼
m10 ; m00
¼ y
The CGM can be normalized [36] into the Normalized Central Geometric Moments (NGM) to make them invariant under translation and scale as shown in Eq. (6).
gpq ¼
l
l
pq ðpþqþ2Þ=2 00
ð6Þ
:
3. Formulation of material characterization as an inverse problem The material characterization problem, in terms of determining the material constants, can be formulated as an inverse problem and reformulated as an optimization problem. 3.1. Inverse problem input The input for the inverse problem is defined in terms of the final shape of a Taylor cylinder. Consider the cylinder, with specific initial diameter and length (D0 ; L0 ), after deforming by impacting against a rigid target at a known initial velocity (v 0 ). See Fig. 2. The cylinder material has an unknown set of constants. Let u~0 ðD0 ; L0 v 0 Þ stand for vector of the measured geometrical coordinates ðx; yÞ of the silhouette of the deformed specimen.
Initialization
Evaluation
Selection
New Population
Random initial population
Fitness for each individual
Parents for next generation
Crossover & Mutation
Fig. 1. Genetic algorithm flowchart.
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L0
D0
v0
Fig. 2. FE model of the Taylor impact test, (left) before and (rigth) after the impact.
3.2. Objective function Now, consider a finite element (FE) model representation of the Taylor event with the same initial geometry (D0 ; L0 ) and impacting at the same velocity (v 0 ). This FE model uses a known trial set of material constants (~ z) that characterize the material. Let u~0 ð~ z; D0 ; L0 v 0 Þ stand for the vector of geometrical coordinates of the computed silhouette of the deformed cylinder. Where the material constants set is: T ~ z ¼ ½r0 ; C; P; b; E; Etan ; l :
ð7Þ
Then, the inverse problem of material characterization can be formulated as follows: Given ~ uð~ z; D0 ; L0 v 0 Þ the coordinates of the silhouette of a cylinder after a Taylor impact test, find ~ z ¼ ½r0 ; C; P; b; E; Etan ; lT the optimal set of material parameters for the kinematic hardening material model. The optimal set of material constants is found when the objective error function, stated as a least-squares minimization problem, proposed in Eq. (8), is minimum.
zÞ ¼ u~0 ð~ z; D0 ; L0 v 0 Þ ~ /err ð~ uð~ z; D0 ; L0 v 0 Þ;
ð8Þ
where k k is the Euclidian norm of the vector. However, this formulation needs to be restated because of the great number of individual measurements of coordinates available for each silhouette and the difficulty to compare them. Then, the optimization problem can be restated in terms of the NGM of the cylinder silhouette coordinates as proposed in Eq. (9).
zÞ ¼ kg~0 ð~ /err ð~ uÞ gð~ uÞk;
ð9Þ
uÞ and gð~ uÞ are the vectors of the CGM of the computed and measured coordinates of the Taylor cylinder silhouwhere g~0 ð~ ettes respectively. This error function is often complicated and contains several local minima, therefore is not well suited to traditional gradient-based optimization techniques. For this reason, a derivative-free optimization process is more appropriate to implement in this particular application. From the wide field of derivative-free optimization techniques the genetic algorithms were selected due to their main features: Genetic algorithms are designed to exploit large highly complicated search spaces efficiently and yet, its implementation is very simple and computer efficient ([33]). The genetic algorithm formulation is explained in next section.
4. Methodology This paper presents a computational methodology for the dynamic characterization of metals from a single Taylor impact test. As shown in Fig. 3, the methodology consists of four basic steps: (1) a single Taylor test at known impact velocity is performed; (2) geometrical moments, up to fourth order, of the specimen’s silhouette are computed; (3) A Taylor test finite element (FE) model with trial material parameters is implemented in order to compare the final shape of the specimen against the experimental test; (4) a genetic algorithm optimization procedure is used to minimize the difference between the geometrical moments computed from the FE model with trial parameters, and geometrical moments from the measured experimental Taylor specimen’s silhouette.
4.1. Finite element model of Taylor test The FE model of the Taylor impact test was implemented using the softwate ANSYS/LS-DYNA employing an explicit solution scheme. Given the axisymmetric nature of the problem, the problem was modeled using axisymetric explicit 2D structural solid elements with six degrees of freedom at each node (PLANE162). The material model used for the Taylor specimen was the Cowper–Symonds material model discussed previously. The impact surface was assumed as a rigid body and the contact frictionless. Fig. 2 shows the FE simulation of the Taylor specimen before and after the impact.
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C. Hernandez et al. / Applied Mathematical Modelling 37 (2013) 4698–4708 INPUT Taylor impact test L0
Measured silhouette Geometric Moments
D0
Genetic Algorithm optimization
v0
OUTPUT Material Parameters
Objective function
Computed silhouette Geometric Moments
GA
FE model Taylor impact test
Fig. 3. Methodology for material characterization from a single Taylor impact test.
4.2. Geometrical moments computation The silhouette of the deformed Taylor specimen can be represented by a function f ðxÞ in which the diameter on each point is a function of the length along the specimen as shown in Fig. 4. This function is obtained by measuring different radius along the specimen length and interpolating a spline function. This spline function is smooth, differentiable, bounded and defined in a finite interval which makes it suitable to compute its geometrical moments. Due to the discrete nature of the FE model and measurements of problem, the expressions on Eqs. (3) and (4) should be approximated by a summation as shown in Eqs. (10) and (11) respectively. Where Dx is the element size in the length direction and N is the number of elements along the specimen’s length. The normalized central geometric moments up to fourth order were computed along with the normalized central geometric moments of the derivative of shape function up to fourth order. The geometric moments of the silhouette derivative are included because they introduce additional information regarding the shape of the specimen and facilitate the comparison between shapes improving the optimization process. The silhouette of the deformed Taylor specimen can be represented by a function in which the diameter on each point is a function of the length along the specimen as shown in Fig. 4, which facilitates the computation of its geometrical moments. Due to the discrete nature of the FE model and the 2D character of the problem, the expressions on Eqs. (3) and (4) should be approximated by a summation as shown in Eqs. (10) and (11), respectively. Where Dx is the element size in the length direction and N is the number of elements along the specimen’s length.
mpq ¼
N X xp f ðxÞqþ1 Dx;
ð10Þ
i¼1
lpq ¼
N X Þq f ðxÞDx: ðx xÞp ðf ðxÞ y i¼1
Fig. 4. Computation of geometrical moments of the deformed Taylor specimen’s silhouette.
ð11Þ
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4.3. Genetic algorithm optimization The optimization of the difference between the experimental and FE geometrical moments was performed by a GA optimization routine. In this optimization problem there are seven genes given by the material parameters that must be determined: ~ z ¼ ½r0 ; C; P; b; E; Etan ; lT . Traditionally, GAs use a binary codification to represent the genes to form a string and then apply the genetic operations. However, other coding types, such as the real codification have been considered to represent the genes. In particular, the real-coded GAs have shown that are more efficient than the binary-coded GAs when optimizing continuous and discrete variables [37]. For this reason real-coded representation of the genes was used is this optimization procedure. The real-coded adaptive range genetic algorithm implemented to characterize the material parameters is shown in Fig. 5 and its operators are described below. 4.3.1. Initialization The first generation of possible solutions is generated by dividing the search space of each parameter uniformly, and then the individuals are generated by the random combination of these parameters. 4.3.2. Evaluation The aptitude of each individual as a solution of the problems is determined by evaluating a fitness function. The fitness function w for every individual is given by Eq. (12), where /err is the objective function defined in Eq. (9).
wð/err Þ ¼
1 : zÞ 1 þ /err ð~
ð12Þ
4.3.3. Selection The selection operator implemented is the roulette wheel selection. This selection operator is based on a fitness proportionate method in which the probability of each individual to be selected as a parent for the next generation is associated to its fitness. Consequently, the highest the fitness of an individual, biggest the probability of being selected as a parent. 4.3.4. New population The new population is generated by applying two operators, crossover and mutation. Three crossover operators are used: heuristic, arithmetic and uniform. These operators are described by Adewuya [38] and the performance for real-coded genetic algorithms noted. The mutation operator used is uniform. Each of the individuals has the exactly equal chance of undergoing mutation. 4.3.5. Range adaptation On the other hand, a range adaptation operator is implemented to prevent the fast convergence to a local minima and loss of diversity. This operator, as shown by Maranon et al. [39], re-generate the entire population every M generations by calculating a new search space range and by generating a new population distributed accordingly to the new parameter range. The new range limits of each parameter is given by the average of the top half of individual in the previous generation, the standard deviation and a parameter k that determines the amplitude of the new range as shown in Eq. (13).
li ¼ xi kr1 ;
ð13Þ
where li are the new range limits for each parameters, xi and individuals of the previous generation for each parameter.
ri are the average and standard deviation of the top half of
5. Numerical analysis The performance and feasibility of the proposed characterization procedure is examined through numerical simulations in which the characterization of 1018 steel was performed. The numerical example consists of a finite elements simulation of a Taylor test specimen using the material constants shown in Table 1. The initial geometry of the specimen is 20 mm length
Every M generations
Range Adaptation New population
Initialization
Evaluation
Selection
New Population
Random initial population
Fitness for each individual
Parents for next generation
Crossover & Mutation
Fig. 5. Real-coded adaptive range genetic algorithm flowchart.
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C. Hernandez et al. / Applied Mathematical Modelling 37 (2013) 4698–4708 Table 1 Material parameters for 1018 steel [40].
l
E 200 GPa
r0
0.27
Etan
310 MPa
C
b
763 MPa
0.3
40 s
1
P
ef
5.0
0.75
Fig. 6. Results of the characterization process for run 4. (—) Taylor specimen shape (- -) predicted specimen shape.
Table 2 Results of characterization process (Five consecutive runs). E [GPa] 200
l [–]
r0 [MPa]
Run
0.27
310
2 3 4 5
195.8 195.5 200.3 199.0 195.7
0.277 0.273 0.280 0.269 0.270
Avg. error
1.4%
(2.1%) (2.2%) (0.2%) (0.5%) (2.1%)
1.6%
(2.6%) (1.1%) (3.7%) (0.4%) (0.0%)
314.4 298.0 289.2 329.0 337.5 5.4%
(1.4%) (3.8%) (6.7%) (6.1%) (8.9%)
Etan [MPa] 763
b [–] 0.30
728.4 702.1 703.0 769.0 857.0
0.340 0.305 0.259 0.337 0.317
6.7%
(4.5%) (8.0%) (7.9%) (0.8%) (12.3%)
9.3%
(13.3%) (1.7%) (13.7%) (12.3%) (5.7%)
C [s1] 40
P [–] 5.0
42.2 43.0 41.2 37.3 42.0
5.05 4.74 4.59 5.12 5.44
5.6%
(5.7%) (7.1%) (3.1%) (6.6%) (5.0%)
Fitness (1.0%) (5.2%) (8.2%) (2.4%) (8.8%)
0.996 0.993 0.995 0.996 0.996
5.12%
and 8 mm diameter. The impact velocity is 250 m/s. After impact the specimen length is 17.6 mm and the undeformed section length is 8.1 mm. The final shape of the Taylor specimen is shown in Fig. 6. The silhouette of the simulated specimen’s deformed shape is used as input parameter in the optimization process. Therefore, the optimization procedure target parameters are shown in Table 1. Normalized central geometrical moments, of the deformed specimen shape, up to fourth order were calculated using Eqs. (4) and (11); and in order to increase the reliance of the optimization procedure on the specimen’s shape, normalized central geometrical moments of the diameter function derivative were also computed. The 30 normalized central moments (15 from the diameter function and 15 from the derivative) were stored in a vector ~ g and used during the optimization in the fitness function. Five runs were performed to determine the efficiency of the characterization procedure. The GA was configured to use 60 individuals and 100 generations. The selection operator used in this work is the roulette wheel combined with elitism strategy. The new individuals are created by using one of three crossover operators implemented: heuristic, arithmetic or uniform. The crossover operator used to create each of the new offspring is selected randomly. The individuals are mutated using a uniform operator with mutation probability of 80%. The range adaptation operator was set to re-generate the entire population every 5 generations with amplitude parameter k of 1.5 standard deviations.
6. Results The feasibility and the effectiveness of the proposed characterization procedure were investigated via numerical simulations. The objective of the characterization example was to determine the Cowper–Symonds material parameters ~ z for the 1018 steel. The procedure used as input the silhouette of a Taylor specimen’s FE model. Five characterization replications were performed and its results are along with the relative estimation error are shown in Table 2. The material parameters were predicted with accuracy above 90% in the five examples. Seven of the eigth parameters were determined with relative error around 5%, while parameter b could only be determined with relative error around 10%. Typical graphical results obtained during this study are shown in Fig. 6. The graphic shows the comparison between the shape of the Taylor specimen
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Fig. 7. Convergence history of parameters for run 4.
C. Hernandez et al. / Applied Mathematical Modelling 37 (2013) 4698–4708
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(—) and the shape of the specimen using the determined parameters (- -). In general, the final shape of the specimen is captured by the proposed characterization procedure, however, small differences between the Taylor specimen silhouettes are observed. These small differences cannot be captured by the geometrical moments used to represent the silhouette. To capture these small differences the order of the geometrical moments used must be increased, expanding the risk of amplifying the noise without the certainty of improving the characterization results. Fig. 7 show the typical convergence history of each of the material parameters as the genetic algorithm evolves, in particular the graphs show the results for run 4 and the evolution of the fitness. The charts show the evolution of each estimated material constant (—), the reference original material constant (-.-), and the search space limits (- -). The main finding of the numerical analysis is that the proposed characterization methodology can determine accurately and consistently the material parameters from a single Taylor impact test. The accuracy was observed in the fact that average global error was maintained below 10%, and the consistency was observed when during consecutive runs the results were kept uniform. Compared with other characterization methods explored in the literature, this proposed method showed three main advantages. The first advantage lies in the simplicity of the method. The use of a single Taylor impact test makes the parameters determination process inexpensive and relatively easy to apply. The second advantage is the simultaneous estimation of the material parameters. Traditional parameter estimation methods uncouple the material model to determine each of the material constants independently ignoring the synergy among parameters. The third advantage is that the characterization process is performed at high strain rates. The materials constants can be determined with impact experiments that undergo strain rates in a broad range from 104 s1 to 106 s1. However, due to the fact that the material parameters are determined using one test at a single strain rate; it is important to identify the strain rate of the particular application in which the material parameters are being use, and perform the experimental Taylor test at this particular average strain rate to obtain the best performance of the material model.
7. Conclusions In this paper a dynamic characterization computational procedure to determine the material parameters from a single Taylor impact test was proposed. The procedure involves the formulation and solution of an inverse problem to determine the Cowper–Symonds parameter of metals using as input the silhouette of a deformed Taylor specimen. Geometrical moments are used to represent the shape of the specimen and a genetic algorithm optimization procedure is used to optimize the set of parameter constants. The proposed characterization technique shows advantages in the material parameters estimation process compared with other characterization strategies: (1) Simplicity and easiness due to the use of a single Taylor test as experimental data. (2) Simultaneous estimation of material parameters that allows observing the interaction among parameters, unlike traditional methods that determine parameters sequentially. (3) Material parameters are determined at high strain rates, in the order of 104 s1 to 106 s1, making this procedure ideal for the development of accurate material models used in numerical simulations of dynamic and impact events. Numerical characterization examples were performed to test the effectiveness of the proposed characterization method. Numerical examples showed that the procedure is capable of capturing the shape of the Taylor specimen and determine the material parameters with accuracy above 90%. Results showed that the algorithm is effective and efficient. More studies need to be conducted to evaluate the performance of the procedure with other materials and experimental Taylor impact tests.
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