Identification of the Parameters of a Concrete Damage Material Model

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ScienceDirect Procedia Engineering 172 (2017) 578 – 585

Modern Building Materials, Structures and Techniques, MBMST 2016

Identification of the Parameters of a Concrete Damage Material Model Petr Krala,*, Petr Hradilb, Jiri Kalac, Filip Hokesd, Martin Huseke a,b,c,d,e

Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Veveri 95, 602 00 Brno, Czech Republic

Abstract Today, the advanced numerical analysis of concrete structures requires the use of nonlinear material models of concrete in order to take into account the nonlinear behavior of concrete within finite element simulations. However, the effective application of nonlinear concrete material models within numerical simulations often becomes problematic because material models often contain parameters whose values are difficult to obtain. Modern computers are advanced enough to solve this problem through the inverse identification of the used material model’s parameters. This inverse identification is based on the combination of optimization methods or procedures with the experimental approach. The aim of this paper is to perform the identification of some parameters of the Karagozian & Case Concrete model - Release III, which is implemented in LS-Dyna software, on the basis of an experimentally-measured loading curve. For this purpose, numerical and experimental approaches were combined with optimization procedures. The loading curve was obtained from a triaxial compression test performed on a concrete cylinder. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MBMST 2016. Peer-review under responsibility of the organizing committee of MBMST 2016 Keywords: Triaxial compression test; concrete cylinder; nonlinear material model of concrete; numerical analysis; inverse identification; loaddisplacement curve.

1. Introduction At present, the continuous and extensive use of concrete for the construction of new building structures is leading to efforts to refine their design through finite element simulations [1,2,3]. However, the drive to design safer and cheaper concrete structures requires the use of advanced numerical analysis. This means that it is necessary to take

* Corresponding author. E-mail address: [email protected]

1877-7058 © 2017 The Authors. 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 the organizing committee of MBMST 2016

doi:10.1016/j.proeng.2017.02.068

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into account the nonlinear behavior of the concrete when designing concrete structures. In the numerical analysis of concrete structures, the nonlinearity of concrete material plays the most important role in terms of the resulting response of the structure. Current innovative finite element solvers, such as ANSYS [4] and LS-Dyna [5], offer various types of nonlinear material models capable of taking concrete material nonlinearity into account. These nonlinear material models of concrete are most often used for the numerical modeling of concrete structures subjected to specific loading, during which the effect of strain rate upon the mechanico-physical properties of concrete is apparent. The previous statement has been demonstrated in publications by authors from around the world, e.g. Kala et al. [6], Kral et al. [7], Wu et al. [8], Yousuf et al. [9], Kala and Husek [10] and Sadiq et al. [11]. However the effective application of nonlinear material models of concrete within numerical simulations is often very problematic because material models very often contain parameters whose values are difficult to obtain. Often they are parameters without physical meaning (i.e. they only have mathematical meaning), or parameters whose numerical values can only be determined on the basis of specific tests carried out on concrete. Today’s computers are sufficiently advanced to be able to solve this problem through the inverse identification of material model parameters. Inverse identification is based on the combination of numerical and experimental approaches with optimization methods or procedures. The most widespread methods in use today for the purpose of the inverse identification of nonlinear material model parameters are methods based on the exercising of artificial neural networks; see Lehky and Novak [12]. A very powerful tool in the area of commercial computing systems is the optiSLang program [13], which offers a broad spectrum of optimization procedures for inverse analysis. The application of this program and information about the optimization procedures and methods it incorporates can be found e.g. in Most [14], Hokes et al. [15], Kala and Kala [16] and Fedorik et al. [17]. The aim of this paper was to perform the inverse identification of some parameters of the Karagozian & Case Concrete model - Release III on the basis of an experimentally-measured loading curve which was obtained from a triaxial compression test performed on a concrete cylinder. This involved the use of a combination of experimental data, a numerical approach via LS-Dyna software (in which the Karagozian & Case Concrete model - Release III is implemented), and optimization procedures implemented in the optiSLang program. 2. Analysis of the experimental data In order to perform the inverse identification of material model parameters, the optiSLang program generally requires the definition of a reference experimentally-measured loading curve. Individual points lying on this curve then serve as output data which are used in calculating the values of the objective function. The inverse identification of parameters of the Karagozian & Case Concrete model - Release III was performed on the basis of a load-displacement curve (see Fig. 1) which was obtained from an experimental investigation published by Joy and Moxley [18]. This curve represents an output of a triaxial compression test carried out on a concrete cylinder with a height of 304.8 mm and a base diameter of 152.4 mm. The ultimate uniaxial compressive strength of the concrete from which the cylinder was manufactured was 45.4 MPa. The confinement was 7 MPa.

Fig. 1. Load-displacement curve from a triaxial compression test.

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It is clear from Fig. 1 that during the compressive loading, the concrete cylinder first exhibited linearly elastic behavior and then elasto-plastic behavior. After exceeding the maximum compressive force (i.e. the maximum triaxial compression load capacity of the concrete cylinder), the concrete began to show signs of compressive strain softening. The compressive strain softening of the concrete began to assert itself as a result of the damage to the concrete cylinder as the ultimate strength of the concrete in triaxial compression was exceeded. However, the total response of the concrete cylinder was very ductile due to the applied confinement (see Fig. 1). 3. Numerical analysis Within the process of the inverse identification performed for this paper, a nonlinear numerical analysis of the triaxial compression test was conducted via the explicit finite element method in LS-Dyna software. The Karagozian & Case Concrete model - Release III was chosen from the material library of LS-Dyna software to provide a description of the nonlinear behavior of the computational model of the cylinder. In order to facilitate the automation of the inverse identification process, all necessary components for the calculation were written in the LS-Dyna keyword file using relevant keywords [19]. This file included the settings of the computational model, the settings of the parameters of the chosen material model, solver settings and other settings necessary for the successful performance of the calculations. 3.1. The Karagozian & Case Concrete model - Release III The Karagozian & Case Concrete model - Release III ([20], [21]), also known as the Concrete Damage material model - Release III, is a newer version of the model known as the Karagozian & Case Concrete model. This material model is defined as a three-invariant model using three shear failure surfaces: an initial yield surface, a maximum shear failure surface and a residual failure surface. These strength surfaces are mutually independent and can be formulated in a generalized form according to Youcai et al. [22] as:

Fi ( p)

a0i 

p a1i  a2i p

(1)

where index i stands for y (initial yield surface), m (maximum shear failure surface) or r (residual failure surface). The variables aji (j = 0, 1, 2) are parameters calibrated from the test data, and p is the pressure, which is dependent on the first invariant of the stress tensor (p = -I1/3). The resulting failure surface is interpolated between the maximum shear failure surface and either the initial yield surface or the residual failure surface according to the following equations: F ( I1 , J 2 , J3 )

r ( J 3 )[K (O )( Fm ( p)  Fy ( p))  Fy ( p)] for O d Om

(2)

F (I1, J 2 , J3 )

r ( J3 )[K(O)( Fm ( p)  Fr ( p))  Fr ( p)] for O ! Om

(3)

where I1 is the first invariant of the stress tensor, J2 and J3 are the invariants (second and third) of the deviatoric stress tensor, λ is the modified effective plastic strain or the internal damage parameter, η(λ) is the function of the internal damage parameter λ and r(J3) is the scale factor in the form of the William-Warnke equation (Chen and Han [23]). The model allows the effect of strain rate, failure and different mechanico-physical properties in compression and tension to be taken into account. It is therefore suitable for concrete modeling. A significant advantage of the model is its ability to generate material parameters. This ability is based exclusively on the specified uniaxial compressive strength. If material parameter generation is used, only 5 parameters need to be defined to ensure the functionality of the material model, though more parameters can be defined if required. Otherwise, the user must define a total of 49 parameters for the material model, and then must

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define the parameters of the equation of state, too. This is very complicated because many parameters have a purely mathematical meaning. For this reason, the developers of the material model recommend the use of material model parameter generation, though they do not exclude the possibility of defining custom parameter values when the resulting response of the material model needs to be refined. For the purposes of this paper the inverse identification was focused on a total of 5 parameters of the material model. These parameters were selected so as to be sufficient for describing the elasto-plastic behavior of the concrete and to permit material model parameter generation. Their description and the units used are given in Table 1. Other parameters were generated or defined by constant values, if necessary. Table 1. Selected parameters of the material model for use in inverse identification process. Parameter

Unit

Description

RO

[Mg/mm ]

Mass density, U.

PR

[-]

Poisson’s ratio, Q.

FT

[MPa]

Uniaxial tensile strength, ft.

A0

[MPa]

Negative value of uniaxial compressive strength, -fc. Necessary for the material parameter generation.

LOCWID

[mm]

Three times the maximum aggregate diameter.

3

3.2. The computational model of the triaxial compression test In a real triaxial compression test described within the publication [18], a concrete cylinder was placed between the pressure plates in the triaxial test chamber of a triaxial apparatus. In order to achieve the objectives of the nonlinear numerical analysis of the triaxial compression test, the boundary conditions were simplified. The cylinder was modeled by itself without the pressure plates, using explicit 3-D structural finite elements. The bottom base of the cylinder was modeled as fixed. This means that zero horizontal and vertical displacements were prescribed for the bottom basal nodes. The top base of the cylinder was modeled as laterally constrained. This means that zero horizontal displacements were prescribed for the top basal nodes. In addition, linearly increased vertical displacements over time were prescribed for the top basal nodes. These displacements simulated the axial compression of the cylinder at a constant velocity. The confinement was applied explicitly by a constant surface pressure over time on the finite element model of the cylinder. The geometry of the cylinder and boundary conditions are depicted in Fig. 2.

Fig. 2. Computational model of the triaxial compression test.

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4. Inverse identification of the material model parameters The inverse identification of parameters of the Karagozian & Case Concrete model - Release III was performed in the user interface of the optiSLang program. The whole process of the inverse identification consisted of three phases. Sensitivity analysis was conducted within the first phase. Global and local optimization was conducted within the second and third phase, respectively. 4.1. Sensitivity analysis When conducting inverse identification using the optiSLang program, sensitivity analysis is employed. The aim is, as the name suggests, to analyze the sensitivity of the input variable data to the output data and consequently reduce the number of input variables in the design vector to the necessary minimum. The range of variability for the individual input variables is also modified in the process of sensitivity analysis. Within the inverse identification detailed in this paper, the design vector included the individual material parameters that were to be identified, and the output data consisted of the points lying on the load-displacement curve. The range of variability for the individual material parameters was obtained based on the test calculations. Sensitivity analysis was performed via the Advanced Latin Hypercube Sampling (ALHS) method. The execution of a total of 200 random realizations of the design vector ensured the satisfactory coverage of the design space. The results of the sensitivity analysis showed that only 3 out of a total of 5 identified material parameters significantly affected the resultant form of the load-displacement curve. Therefore, the original design vector:

Xoriginal

^RO, PR, FT , A0, LOCWID`

T

(4)

could be reduced for the subsequent optimization process into the form:

Xreduced

^PR, A0, LOCWID`

T

(5)

This reduction was used for the subsequent global and local optimization because it enabled computation time and disk space to be saved without any significant effect on the achieved results. The values of the pre-optimized parameters from the best sensitivity analysis realization are documented in Tab. 2. The value of the objective function (ERROR_NORM) from the best sensitivity analysis realization is also documented in Tab. 2. This is the lowest value obtained from all realizations. 4.2. Global and local optimization Generally, the aim of conducting optimization in the optiSLang program is to find a configuration of the values of optimized parameters that caused the value of the target function to be minimized or maximized. For the purposes of the global and local optimization in this paper, the target objective function was defined as:

¦ y n

ERROR _ NORM

s ,i

 yr ,i

2

(6)

i 1

where ys,i were the force values obtained from the appropriate numerically-simulated load-displacement curve and yr,i were the force values corresponding to the experimentally-measured load-displacement curve obtained from the triaxial compression test. The objective function was minimized (ERROR_NORM o min). This means that material parameter values were sought that caused the load-displacement curve obtained from the numerical simulation to show the smallest deviation from the experimentally-measured loading curve. The reduced design vector was used for the optimization process, as already mentioned.

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The global optimization of the material parameters was performed using an optimization procedure known as the Evolutionary Algorithm (EA). This method is a combination of the Genetic Algorithm (GA) with Evolution Strategies (ES). The 10 best sensitivity analysis realizations were used as a start point for the global optimization process. The values of the identified parameters from the best Evolutionary Algorithm generation are documented in Tab. 2, including the minimized value of the objective function. Table 2. Resulting material model parameter values from the inverse identification process. Parameter

Unit

Sensitivity analysis

Evolutionary Algorithm

Downhill Simplex

RO

[Mg/mm3]

2.165˜10-9

2.4˜10-9

2.4˜10-9

PR

[-]

0.1657

0.1619

0.1619

FT

[MPa]

3.3579

3.36

3.36

A0

[MPa]

-43.485

-43.663

-43.663

LOCWID

[mm]

70.98

73.3654

73.3654

16198.5

14525.4

14525.4

ERROR_NORM

2

[kN ]

The subsequent local optimization was conducted via the Downhill Simplex (DS) method, which is suitable for the optimization of a small number of parameters. The best Evolutionary Algorithm generation of the design vector was used as a start point for this process. However, the Downhill Simplex method did not provide further results enhancement, and therefore the results of this method are the same as those obtained from the Evolutionary Algorithm (see Tab. 2). A comparison of the load-displacement curve for the optimum material parameter values obtained from the Evolutionary Algorithm with the experimentally-measured loading curve is shown in Fig. 3. The load-displacement curve obtained from the numerical simulation, in which only the material parameter generation based on the uniaxial compressive strength (45.4 MPa) was used without the process of inverse identification, is also shown in Fig. 3 for comparison. It is clear from Fig. 3 that the inverse identification process was necessary in this case to obtain a satisfactory approximation of the experimental data.

Fig. 3. Comparison of the load-displacement curves for the triaxial compression test.

5. Conclusion In this paper, the parameters of the Karagozian & Case Concrete model - Release III were identified via optimization procedures implemented in the optiSLang program. This inverse identification was performed on the basis of an experimentally-measured loading curve obtained from the triaxial compression test. The results of the inverse identification showed that the used material model of concrete is able to describe the behavior of the given real concrete in triaxial compression very well provided that the values of its absolutely necessary parameters are

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appropriately specified. Furthermore, the results of the inverse identification showed that only parameters which influence compressive loading had to be identified. Other parameters could be filtered out of the design vector. For the identification of these parameters it is necessary to use experimentally-obtained curves from different tests. The approximation of the experimentally-measured loading curve by the load-displacement curve obtained from the numerical simulation, in which the optimum material parameter values were used, was quite accurate. However, a more accurate approximation could be achieved by, e.g. using different optimization procedures, or by increasing the number of generations of the design vector. These options form the basis for further research by the authors. Acknowledgement This paper was created with the financial support of project GACR 14-25320S “Aspects of the use of complex nonlinear material models” provided by the Czech Science Foundation and also with the support of project FAST-J16-3744 “Optimization of the parameters of nonlinear concrete material models for explicit dynamics” of the Specific University Research of Brno University of Technology. References [1] J. Kala, Z. Kala, The Interaction of Local Buckling and Stability Loss of a Thin-Walled Column under Compression, in: T.E. Simos, G. Psihoyios, C. Tsitouras et al (Eds.), AIP Conference Proceedings 1479 (2012) 2074-2077. [2] J. Kala, V. Salajka, P. Hradil, Investigation of eigenvalue problem of water tower construction interacting with fluid, Journal of Vibroengineering 14 (2012) 1151-1159. [3] J. Kala, V. Salajka, P. Hradil, Calculation of Timber Outlook Tower with Influence of Behavior of “Steel-Timber” Connection, in: G. 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