Issue 39

Frattura ed Integrità Strutturale, 39 (2017); International Journal of the Italian Group of Fracture

Table of Contents

J. Eliáš On adaptive refinements in discrete probabilistic fracture models …………….…………...………. 1 F. Hokes, J. Kala Selecting the objective function during the inverse identification of the parameters of a material model of concrete ……………………………………………………………………….……...… 7 J. Klon, V. Veselý Modelling of size and shape of damage zone in quasi-brittle notched specimens – analytical approach based on fracture-mechanical evaluation of loading curves …………….………………………… 17 P. Konecny, P. Lehner Effect of cracking and randomness of inputs on corrosion initiation of reinforced concrete bridge decks exposed to chlorides …………….……........................................................................................... 29 P. Král, P. Hradil, J. Kala Inverse identification of the material parameters of a nonlinear concrete constitutive model based on the triaxial compression strength testing ………………………………………………………… 38 J. Labudkova, R. Cajka Numerical analyses of interaction of steel-fibre reinforced concrete slab model with subsoil ………….… 47 A. Lokaj, K. Klajmonová Comparison of behaviour of laterally loaded round and squared timber bolted joints ............................... 56 J. Flodr, O. Sucharda, D. Mikolasek, P. Parenica Experiment and numerical modeling suspended ceiling with identification of working diagram material 62 J. Navrátil, M. Číhal, J. Kabeláč, R. Štefan Nonlinear analysis of reinforced and composite columns in fire …………………………….…... 72 V. Salajka, J. Klouda, P. Hradil Numerical simulations of tests masonry walls from ceramic block using a detailed finite element model . 88 S. Seitl, R. Diego Liedo Numerical study and pilot evaluation of experimental data measured on specimen loaded by bending and wedge splitting forces ……........................................................................................................... 100

I

Fracture and Structural Integrity, 39 (2017); ISSN 1971-9883

S. Seitl, T. Thienpont, W. De Corte Fatigue crack behaviour: comparing three-point bend test and wedge splitting test data on vibrated concrete using Paris' law ………………………………………………………………...… 110 S. Seitl, V. Viszlay Modified compact tension specimen for experiments on cement based materials: comparison of calibration curves from 2D and 3D numerical solutions .………......................................................................... 118 J. Sobek, P. Frantík, V. Veselý Analysis of accuracy of Williams series approximation of stress field in cracked body – influence of area of interest around crack-tip on multi-parameter regression performance …………………………… 129 M. Krejsa, L. Koubova, J. Flodr, J. Protivinsky, Q. T. Nguyen Probabilistic prediction of fatigue damage based on linear fracture mechanics .......................................... 143 M. Muñiz Calvente, S. Blasón, A. Fernández Canteli, A. de Jesús, J. Correia A probabilistic approach for multiaxial fatigue criteria ........................................................................ 160 M. Shariati, M. Mahdizadeh Rokhi, H. Rayegan Investigation of stress intensity factor for internal cracks in FG cylinders under static and dynamic loading ………………………………………………………………………………..... 166 H. Xiao, L. Qing, C. Hongkai, T. Hongmei, W. Linfeng Experimental analysis on physical and mechanical properties of thermal shock damage of granite …….. 181 M. A. Lepore, M. Perrella From test data to FE code: a straightforward strategy for modelling the structural bonding interface … 191 A. Risitano, D. Corallo, E. Guglielmino, G. Risitano, L. Scappaticci Fatigue assessment by energy approach during tensile tests on AISI 304 steel ………......………… 202 S. K. Kudari, K. G. Kodancha 3D Stress intensity factor and T-stresses (T 11 and T 33 ) formulations for a Compact Tension specimen . 216 M. Romano, I. Ehrlich, N. Gebbeken Parametric characterization of a mesomechanic kinematic caused by ondulation in fabric reinforced composites: analytical and numerical investigations ………………………………………….… 226 M. A. Tashkinov Modelling of fracture processes in laminate composite plates with embedded delamination ........................ 248 O. Daghfas, A. Znaidi, A. Ben Mohamed, R. Nasri Experimental and numerical study on mechanical properties of aluminum alloy under uniaxial tensile test .................................................................................................................................................... 263 S. Doddamani, M. Kaleemulla Experimental investigation on fracture toughness of Al6061–graphite by using Circumferential Notched Tensile Specimens .............................................................................................................................. 274 S. Harzallah, M. Chabaat, K. Chabane Numerical study of eddy current by Finite Element Method for cracks detection in structures .................. 282

II

Frattura ed Integrità Strutturale, 39 (2017); International Journal of the Italian Group of Fracture

Editor-in-Chief Francesco Iacoviello

(Università di Cassino e del Lazio Meridionale, Italy)

Associate Editors Alfredo Navarro

(Escuela Superior de Ingenieros, Universidad de Sevilla, Spain) (Ecole Nationale Supérieure d'Arts et Métiers, Paris, France)

Thierry Palin-Luc

Luca Susmel John Yates

(University of Sheffield, UK) (University of Manchester, UK)

Guest Editors (Modelling in Mechanics) Petr Konecny

(VŠB - Technical University of Ostrava, Czech Republic) (VŠB - Technical University of Ostrava, Czech Republic)

Martin Krejsa

Advisory Editorial Board Harm Askes

(University of Sheffield, Italy) (Tel Aviv University, Israel) (Politecnico di Torino, Italy) (Università di Parma, Italy) (Politecnico di Torino, Italy)

Leslie Banks-Sills Alberto Carpinteri Andrea Carpinteri Emmanuel Gdoutos Youshi Hong M. Neil James Gary Marquis Ashok Saxena Darrell F. Socie Shouwen Yu Ramesh Talreja David Taylor Robert O. Ritchie Cetin Morris Sonsino Donato Firrao

(Democritus University of Thrace, Greece ) (Chinese Academy of Sciences, China)

(University of Plymouth, UK)

(Helsinki University of Technology, Finland)

(University of California, USA)

(Galgotias University, Greater Noida, UP, India; University of Arkansas, USA)

(University of Illinois at Urbana-Champaign, USA)

(Tsinghua University, China) (Fraunhofer LBF, Germany) (Texas A&M University, USA) (University of Dublin, Ireland)

Editorial Board Stefano Beretta

(Politecnico di Milano, Italy)

Filippo Berto Nicola Bonora

(Norwegian University of Science and Technology, Norway) (Università di Cassino e del Lazio Meridionale, Italy)

Elisabeth Bowman

(University of Sheffield) (Università di Parma, Italy) (Politecnico di Torino, Italy) (EADS, Munich, Germany) (EDAM MIT, Portugal)

Luca Collini

Mauro Corrado

Claudio Dalle Donne Manuel de Freitas Vittorio Di Cocco Giuseppe Ferro Tommaso Ghidini Eugenio Giner Paolo Lonetti Carmine Maletta Liviu Marsavina Daniele Dini

(Università di Cassino e del Lazio Meridionale, Italy)

(Imperial College, UK)

(Politecnico di Torino, Italy)

(European Space Agency - ESA-ESRIN) (Universitat Politecnica de Valencia, Spain)

(Università della Calabria, Italy) (Università della Calabria, Italy) (University of Timisoara, Romania) (University of Porto, Portugal)

Lucas Filipe Martins da Silva

III

Fracture and Structural Integrity, 39 (2017); ISSN 1971-9883

Hisao Matsunaga Mahmoud Mostafavi

(Kyushu University, Japan) (University of Sheffield, UK)

Marco Paggi Oleg Plekhov

(IMT Institute for Advanced Studies Lucca, Italy)

(Russian Academy of Sciences, Ural Section, Moscow Russian Federation)

Alessandro Pirondi

(Università di Parma, Italy)

Luis Reis

(Instituto Superior Técnico, Portugal)

Giacomo Risitano Roberto Roberti

(Università di Messina, Italy) (Università di Brescia, Italy) (Università di Bologna, Italy) (University of Belgrade, Serbia) (Università di Parma, Italy) (Università di Parma, Italy)

Marco Savoia

Aleksandar Sedmak Andrea Spagnoli Sabrina Vantadori Charles V. White

(Kettering University, Michigan,USA)

IV

Frattura ed Integrità Strutturale, 39 (2017); International Journal of the Italian Group of Fracture

Journal description and aims Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is the official Journal of the Italian Group of Fracture. It is an open-access Journal published on-line every three months (July, October, January, April). Frattura ed Integrità Strutturale encompasses the broad topic of structural integrity, which is based on the mechanics of fatigue and fracture, and is concerned with the reliability and effectiveness of structural components. The aim of the Journal is to promote works and researches on fracture phenomena, as well as the development of new materials and new standards for structural integrity assessment. The Journal is interdisciplinary and accepts contributions from engineers, metallurgists, materials scientists, physicists, chemists, and mathematicians. Contributions Frattura ed Integrità Strutturale is a medium for rapid dissemination of original analytical, numerical and experimental contributions on fracture mechanics and structural integrity. Research works which provide improved understanding of the fracture behaviour of conventional and innovative engineering material systems are welcome. Technical notes, letters and review papers may also be accepted depending on their quality. Special issues containing full-length papers presented during selected conferences or symposia are also solicited by the Editorial Board. Manuscript submission Manuscripts have to be written using a standard word file without any specific format and submitted via e-mail to gruppofrattura@gmail.com. Papers should be written in English. A confirmation of reception will be sent within 48 hours. The review and the on-line publication process will be concluded within three months from the date of submission. Peer review process Frattura ed Integrità Strutturale adopts a single blind reviewing procedure. The Editor in Chief receives the manuscript and, considering the paper’s main topics, the paper is remitted to a panel of referees involved in those research areas. They can be either external or members of the Editorial Board. Each paper is reviewed by two referees. After evaluation, the referees produce reports about the paper, by which the paper can be: a) accepted without modifications; the Editor in Chief forwards to the corresponding author the result of the reviewing process and the paper is directly submitted to the publishing procedure; b) accepted with minor modifications or corrections (a second review process of the modified paper is not mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. c) accepted with major modifications or corrections (a second review process of the modified paper is mandatory); the Editor in Chief returns the manuscript to the corresponding author, together with the referees’ reports and all the suggestions, recommendations and comments therein. d) rejected. The final decision concerning the papers publication belongs to the Editor in Chief and to the Associate Editors. The reviewing process is usually completed within three months. The paper is published in the first issue that is available after the end of the reviewing process.

Publisher Gruppo Italiano Frattura (IGF) http://www.gruppofrattura.it ISSN 1971-8993 Reg. Trib. di Cassino n. 729/07, 30/07/2007

Frattura ed Integrità Strutturale (Fracture and Structural Integrity) is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0)

V

Fracture and Structural Integrity, 39 (2017); ISSN 1971-9883

10 years!

D

ear friend, ten years ago, in 2007, the IGF Ex-Co decided to publish an open access on line journal… Frattura ed Integrità Strutturale was born! Starting from a few papers per issue, many of them in italian, now Frattura ed Integrità Strutturale is a really well known international journal. Hundreds of papers with hundreds of authors from all over the world were published in the last years and the journal indexing and evaluations are absolutely of the highest level. This great result is only due to an enthusiast group of friends that served (and serves!) the journal in different roles (Associate Editors, Guest Editors, Advisory Editorial Board members, Editorial Board members, Reviewers) and to a long list of authors that in these years supported our journal with their submissions ... THANK YOU!!! In these years, we tried to do our best activating some innovative services. Among them: - Papers published both in pdf and in a browsable format; - The possibility to embed and discuss videos in the published papers; - The possibility to publish papers with audioslides embedded; In this issue, we will offer you a new service. On the top-left side of the first page of each paper, you will find an “audio” symbol. If you will select this symbol, a voice will read title, abstract, keywords and conclusions of the paper. Hoping that this new service will be useful, thank you so much for supporting us,

Francesco Iacoviello F&IS Chief Editor

P.S. Last news."Frattura ed Integrità Strutturale" now has a CiteScore: 0.72 (2015)

2007

2017

VI

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

Focussed on Modelling in Mechanics

On adaptive refinements in discrete probabilistic fracture models

J. Eliáš Brno University of Technology, Faculty of Civil Engineering, Veveří 331/95, Brno, 60200, Czech Republic elias.j@fce.vutbr.cz

A BSTRACT . The possibility to adaptively change discretization density is a well acknowledged and used feature of many continuum models. It is employed to save computational time and increase solution accuracy. Recently, adaptivity has been introduced also for discrete particle models. This contribution applies adaptive technique in probabilistic discrete modelling where material properties are varying in space according to a random field. The random field discretization is adaptively refined hand in hand with the model geometry. K EYWORDS . Adaptivity; Discrete model; Probability; Random field.

Citation: Eliáš, J., On adaptive refinements in discrete probabilistic fracture models, Frattura ed Integrità Strutturale, 39 (2017) 1-6.

Received: 11.07.2016 Accepted: 12.09.2016 Published: 01.01.2017

Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

I NTRODUCTION

T

he adaptivity of model geometry has been originally developed for elastic problems [1,2] and later applied also in inelastic problems with localization [3,4]. The classical rigorous approach involves an error estimation, remeshing criterion, mesh re-generation and transfer of variables onto the new mesh. Recently, the adaptive concept was applied also in discrete modelling [5]. The goal of this work is to extend it for probabilistic discrete models. Discrete models represent the material via collection of interconnected rigid bodies organized into a net structure. There are several versions of discrete models developed and used for many purposes. In case of simulating fracture in concrete, the lattice models are often employed [6-8]. These models represent the concrete meso-structure by projecting it onto the independently generated lattice. They are excellent in describing fracture phenomena, but applicable only for small laboratory specimens due to their extreme computational demands. Another group of discrete meso-level modelling approaches, sometimes called particle models, generates the network geometry directly according to the meso-structure of concrete [9,10]; typically one node for each mineral aggregate. We focus here on the latter group with geometry generated via Voronoi tessellation [11-14]. Though some reduction of computational cost in particle models is achieved when compared to the lattice models, further reduction would be desirable. It can be done by adaptive construction of the discrete geometry as described in [5]. Availability of adaptive refinement allows starting simulation with coarse discretization and refining it adaptively during the simulation run only in areas where needed.

1

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

In some applications of fracture simulations, it might be important to consider additional material randomness (besides the one covered by the random location of nodes in the discrete model) usually represented by a random field [15-18]. An extension of the discrete model by fluctuation of material parameters according to a random field was developed in [18,19]. In this contribution, the adaptive concept is extended for such probabilistic discrete fracture models.

P ROBABILISTIC DISCRETE MODEL

T

he model uses random geometry to avoid directional bias that occurs in any regular structure. Domain of the modeled body is filled with nuclei with randomly generated positions. These nuclei are added sequentially with restricted minimal distance l min . The parameter l min controls size of the discrete bodies and therefore it should correspond to the size of heterogeneities in the material. In concrete, this is typically a size of the mineral aggregates. Each of the nuclei will serve as one model node with associated six degrees of freedom, three translational and three rotational. The connectivity of the nodes is given by Delaunay triangulation. Dual diagram of Delaunay triangulation called Voronoi tessellation then creates geometry of the rigid bodies. Rigid bodies have common contact facets, which are perpendicular to their connections because of the Voronoi tessellation properties. There is a complex damage-mechanics based constitutive law used at the facets. Its deterministic version has been adapted from [9], where it is also described in detail. The main material parameters for fracture behavior are tensile strength, f t , and tensile fracture energy, G F . The probabilistic extension of the model is elucidated in [19,20]. Here, only brief description of the probabilistic part is given. Both the tensile strength and fracture energy in tension are assumed to be governed by single random field H with mean value 1 and probabilistic distribution with Gaussian core and Weibull left tail. The correlation structure of the random field is given by square exponential function with single parameter, l ρ , called the correlation length. The strength and fracture energy of every model contact with centroid c are given by     f f H t t  c c (1)     G G H 2 F F      c c with X being the mean value of the material parameter X . The square in the equation for fracture energy is added to preserve constant material characteristic length [20]. In the adaptive model, new contacts are created after every refinement. Therefore, the random field values at the new contact centers must be generated after every refinement. This is effectively done using kriging. Initially, standard Gaussian random field realizations ( H ˆ ) are generated on points arranged in a regular grid with spacing l ρ /4. Random field value at point c is then estimated using the optimal linear estimation method [21]

K



k    c 1

H ˆ

 

T k k cg

ψ C

(2)

k 

and finally standard Gaussian field is transformed onto the Weibull-Gauss field ( H H ˆ  ) using isoprobabilistic transformation. Vector ξ collects realizations of K independent standard Gaussian variables, λ and ψ are K eigenvalues and eigenvectors of the grid covariance matrix and cg C is the covariance vector between the grid points and point c .

A DAPTIVITY

O

nly brief description of the adaptive concept in deterministic model is given. Deep elucidation is provided in [5]. The refinement criterion is intuitive. It is based on an average stress in the rigid bodies calculated using the fabric stress tensor. For rigid body associated with node i, the average stress components st  are

2

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

st V 1   

    j j

F c

(3)

s

t

j

where j runs over all nodes in contact with node i , F is a vector of contact force, c is the centroid of the contact facet and V is a volume of the i -th rigid body. The Mazar's equivalent stress serves as measure of the stress level, σ eq .

Figure 1 : Adaptive refinement of discretization in steps; a) schematic explanation; b)-g) application to a 2D model.

In probabilistic model, the contacts have random strength. Assuming that the random field does not change too much within one discrete body of the model ( l ρ < l min ), it is reasonable to estimate strength of hypothetical newly created contacts within the i -th coarse discrete body by strength at node i at coordinates i x . After every solution step, average stress tensors in all rigid bodies belonging to coarse discretization are evaluated and the Mazar‘s equivalent stresses are calculated. The refinement takes place whenever the equivalent stress exceeds chosen strength level γ



eq





(4)

  i x

f

t

The node associated with the rigid body satisfying Eq. (4) serves as a center of the refinement sphere. The safe value of parameter γ was determined as 0.7, i.e. whenever equivalent stress reaches 70% of the tensile strength, the refinement takes place. The refinement is sketched in Fig. 1. All the nuclei inside the refinement sphere that does not belong to the fine discretization are removed. New nuclei are added into the refinement sphere according to the sequential algorithm described in the previous section. The parameter l min controlling the discretization density changes based on two additional length parameters, r f and r c . The linear transition from coarse ( l min =l c ) to fine ( l min =l f ) discretization is included within the circular ring of outer (inner) radius r c ( r f ) in order to minimize the shape distortion of the bodies. If the transitional regime is omitted, the sharp change in discretization density would produce significantly elongated body shapes inducing directional bias and anisotropy.

3

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

N UMERICAL EXAMPLE

P

erformance of the proposed adaptive algorithm is demonstrated on simulation of four-point bending test with incorporated material randomness. The computer code used for calculation is an in-house software. The beam geometry is shown in Fig 2. The deterministic model parameters were taken from simulation of experimental series in three-point bending [22]. The average tensile strength is f t =2.2 MPa, fracture energy in tension is G F =35 J/m 2 and elastic modulus is 60 GPa. All these parameters are applied on the meso-level, they are not equal to the corresponding macroscopic properties of the model. The adaptive algorithm uses the following parameters: r f =60 mm, r c =120 mm, l f =10 mm and l c =30 mm. The parameters of the probabilistic extension are arbitrarily chosen according to [19]. The correlation length and the coefficient of variation of the random field is 80 mm and 0.25, respectively. Three model types are used: (i) the fine model, that uses fine discretization everywhere from the beginning; (ii) the coarse model, that uses coarse discretization all the time; and (iii) the adaptive model, that starts with coarse discretization and refines it adaptively. Fig. 3 shows on the left-hand side identical responses of one simulation using the fine model and one simulation using the adaptive model with the same refined meso-structure and also the same random field realization. The resulting crack patterns as well as the random field applied are shown in Fig. 4.

Figure 2 : Dimensions of the simulated beam loaded in four-point bending.

Figure 3 : Left: Load-displacement response of one four-point-bending test simulation using the fine model and the adaptive model with the same refined meso-structure. Right: Average response of 30 simulation of four-point-bending test.

All three model types were then compared statistically. The same 30 realizations of the random field were used for every model type. The average responses together with standard deviations are shown in Fig. 3 on the right hand side. The fine and the adaptive model exhibit the same behavior while the coarse model deviates from them. In average, the computational time consumed by the adaptive model was only 47% of the time consumed by the fine model.

4

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

Figure 4 : Damage patterns and random field discretizations developed during the simulation of four-point bending test for the adaptive and fine model.

C ONCLUSIONS

T

he probabilistic discrete model has been extended by an adaptive technique that allows significant reduction of the computational time with no effect on the obtained results. The probabilistic model had both fracture energy and tensile strength assigned according to the random field. The random field discretization was adaptively refined on the run hand to hand with the discretization of the model. The adaptive algorithm was verified by simulating four-point bending test. Usage of the adaptive concept is limited to the specific types of the discrete models that have elastic behavior independent on discretization density. Moreover, the presented concept is available only for static models. In dynamics, the translations and rotations and their first and second order derivatives cannot be computed from scratch and needs to be somehow estimated from replaced coarse mesh via some transfer algorithm.

A CKNOWLEDGEMENT

T

he financial support provided by the Ministry of Education, Youth and Sports of the Czech Republic under the project LO1408 ‘‘AdMaS UP - advanced Materials, Structures and Technologies’’ under ‘‘National Sustainability Programme I’’ is gratefully acknowledged.

5

J. Eliáš, Frattura ed Integrità Strutturale, 39 (2017) 1-6; DOI: 10.3221/IGF-ESIS.39.01

R EFERENCES

[1] Babuška, I., Rheinboldt, W.C., A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Eng., 12 (1978) 1597-1615. DOI: 10.1002/nme.1620121010. [2] Zienkiewicz, O.C., Zhu, J.Z., A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Eng., 24 (1978) 337-357. DOI: 10.1002/nme.1620240206. [3] Selman, A., Hinton, E., Bičanič, N., Adaptive mesh refinement for localised phenomena. Comput. Struct., 63 (1997) 475-495. DOI: 10.1016/S0045-7949(96)00372-0. [4] Patzák, B., Jirásek, M., Adaptive resolution of localized damage in quasi-brittle materials. J. Eng. Mech.-ASCE, 130 (2004) 720-732. DOI: 10.1016/S0045-7949(96)00372-0. [5] Eliáš, J., Adaptive technique for discrete models of fracture. Int. J. Solids Struct., accepted for publication. DOI: 10.1016/j.ijsolstr.2016.09.008. [6] Man, H.-K., van Mier, J.G.M., Damage distribution and size effect in numerical concrete from lattice analyses. Cement Concrete Comp., 33 (2011), 867-880. DOI: 10.1016/j.cemconcomp.2011.01.008. [7] Sands, C.M., An irregular lattice model to simulate crack paths in bonded granular assemblies. Comput. Struct., 162 (2016) 91-101. DOI: 10.1016/j.compstruc.2015.09.006. [8] Eliáš, J., Stang, H., Lattice Modeling of Aggregate Interlocking in Concrete. Int. J. Fracture, 175 (2012) 1-11. DOI: 10.1007/s10704-012-9677-3. [9] Cusatis, G., Cedolin, L., Two-scale study of concrete fracturing behavior. Eng. Fract. Mech., 74 (2007) 3-17. DOI: 10.1016/j.compstruc.2015.09.006. [10] Cusatis, G., Pelessone, D., Mencarelli, A., Lattice discrete particle model (LDPM) for failure behavior of concrete. I: Theory. Cement Concrete Comp., 33 (2011), 881-890. DOI: 10.1016/j.cemconcomp.2011.02.011. [11] Eliáš, J., Le, J.-L., Modeling of mode-I fatigue crack growth in quasibrittle structures under cyclic compression. Eng. Fract. Mech., 96 (2012) 26-36. DOI: 10.1016/j.engfracmech.2012.06.019. [12] Gedik, Y.H., Nakamura, H, Yamamoto, Y., Kuneida, M., Evaluation of three-dimensional effects in short deep beams using a rigid-body-spring-model. Cement Concrete Comp., 33 (2011) 978-991. DOI: 10.1016/j.cemconcomp.2011.06.004. [13] Veselý, V., Frantík, P., Vodák, O., Keršner, Z., Localization of Propagation of Failure in Concrete Specimens Assessed by Means of Acoustic and Electromagnetic Emission and Numerical Simulations, Transactions of the VŠB – Technical University of Ostrava, Civil Engineering Series, 11 (2011) 1213-1962. DOI: 10.2478/v10160-011-0036-5. [14] Frantík, P., Veselý, V., Keršner, Z., Parallelization of lattice modelling for estimation of fracture process zone extent in cementitious composites. Adv. Eng. Softw., 60-61 (2013) 48-57. DOI: 10.1016/j.advengsoft.2012.11.020. [15] Georgioudakis, M., Stefanou, G., Papadrakakis, M., Stochastic failure analysis of structures with softening materials. Eng. Struct., 61 (2014) 13-21. DOI: 10.1016/j.engstruct.2014.01.002. [16] Grassl, P., Bažant, Z.P., Random lattice-particle simulation of statistical size effect in quasibrittle structures failing at crack initiation. J. Eng. Mech.-ASCE, 135 (2009) 85-92. DOI: 10.1061/(ASCE)0733-9399(2009)135:2(85). [17] Vořechovský, M., Sadílek, V., Computational modeling of size effects in concrete specimens under uniaxial tension. Int. J. Fracture, 154 (2008) 27-49. DOI: 10.1007/s10704-009-9316-9. [18] Vořechovská, D., Vořechovský, M., Analytical and Numerical Approaches to Modelling of Reinforcement Corrosion in Concrete, Transactions of the VŠB – Technical University of Ostrava, Civil Engineering Series, 14 (2014) 20-30. DOI: 10.2478/tvsb-2014-0003. [19] Eliáš, J., Vořechovský, M., Skoček, J., Bažant, Z.P., Stochastic discrete meso-scale simulations of concrete fracture: comparison to experimental data. Eng. Fract. Mech., 135 (2015) 1-16. DOI: 10.1016/j.engfracmech.2015.01.004. [20] Eliáš, J., Kaděrová, J., Vořechovský, M., Interplay of probabilistic and deterministic internal lengths in simulations of concrete fracture. in: Saouma, V., Bolander, J., Landis, E. (Eds.) 9th International Conference on Fracture Mechanics of Concrete Structures, Berkley, USA, (2016). DOI: 10.21012/FC9.155. [21] Li, C.-C., Der Kiureghian, A., Optimal discretization of random fields. J. Eng. Mech.-ASCE, 119 (1993) 1136-1154. DOI: 10.1061/(ASCE)0733-9399(1993)119:6(1136). [22] Gregoire, D., Rojas-Solano, L.B., Pijaudier-Cabot, G., Failure and size effect for notched and unnotched concrete beams. Int. J. Numer. Anal. Met., 37 (2013) 1434-1452. DOI: 10.1002/nag.2180.

6

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

Focussed on Modelling in Mechanics

Selecting the objective function during the inverse identification of the parameters of a material model of concrete

F. Hokes, J. Kala Faculty of Civil Engineering, Brno University of Technology, Veveri 331/95 Brno 60200

hokes.f@fce.vutbr.cz, http://www.fce.vutbr.cz kala.j@fce.vutbr.cz, http://www.fce.vutbr.cz

A BSTRACT . Selecting the correct objective function is the critical precondition for a successful optimization task. The validity of this condition is also required when optimization algorithms are needed for the inverse identification of the unknown parameters of nonlinear material models of concrete, where experimentally measured load-displacement curves can be conveniently applied. In such cases, the objective function expressions can be formulated as the difference between the functional values of the curves or via comparing the characteristic features, which comprise the area under the curve and also the maximum functional value. The proposed article brings a study of the influence of the different formulations of the objectives functions to achieving optimum in the inverse analysis using genetic algorithm. The numerical part of the study was performed in the ANSYS computational system with use of multiPlas library of elasto-plastic material models from which the model based on formulations of Menetrey and Willam was chosen. K EYWORDS . Identification; Objective function; RMSE; Optimization; Sensitivity analysis.

Citation: Hokes, F., Kala, J., Selecting the objective function during the inverse identification of the parameters of a material model of concrete, Frattura ed Integrità Strutturale, 39 (2017) 7-16.

Received: 11.07.2016 Accepted: 21.09.2016 Published: 01.01.2017

Copyright: © 2017 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

I NTRODUCTION

he application of advanced nonlinear constitutive models of building materials can be described as the approximation of mathematical modelling methods to the real behaviour of structures. Such efforts are, however, often complicated by the existence of a wide set of input parameters for such nonlinear models. Interest in the phenomenon of nonlinear behavior of concrete is in order of a wide range of use of this material in scope of many researchers. However, the construction of a correct constitutive relationship which is able to express this nonlinear behaviour for various types of loading appears to be problematic [1]. One of the basic problems which arise when formulating a material model for concrete is the different responses of the material to tensile and compressive load [2]. For this reason, several approaches are used for the mathematical description of the behaviour of concrete. One of these approaches involves the use of theory of plasticity [3]. Applications of theory of plasticity to the description of the T

7

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

behaviour of plain concrete can be found in the work of authors in [4], Willam and Warnke [5], Bazant [6], Dragon and Mroz [7], Schreyer [8], Chen and Buykozturk [9], Onate [10], Pramono and Willam [11], Etse and Willam [12], Menetrey and Willam [13], and Grassl [14]. The use of pure plasticity theory is not sufficient due to the gradual decrease in the stiffness of concrete due to the occurrence of cracks [1]. This problem can be removed when damage theory is used, i.e. by using an adequate damage model. However, as Grassl claims [15], independent damage models are not sufficient when the description of irreversible deformations and the inelastic volumetric expansion of concrete is required. Despite the above-mentioned limitations of both approaches, there are advantages to using both of them in mutual combination, and they can be combined further with other approaches formulated within the framework of nonlinear fracture mechanics. However, the use of combined material models results in a problem in real life in the form of the large amount of parameters which need to be known for the selected material model before the launch of the numerical simulation itself. Unfortunately, not all data regarding these parameters, which can be both mechanico-physical and fracture-mechanical in nature, may be available in advance. This problems can be resolved with use of other advanced mathematical approaches like optimization analysis. The varied spectrum of possibilities characterizing the use of optimization methods includes, among other options, the above mentioned inverse identification of the unknown parameters of the nonlinear material models utilized in numerical analyses performed with the finite element method. The optimization algorithms exploited in the inverse analysis of unknown material parameters, described within references [16, 17], constitute a counterpart to methods based on the training of artificial neural networks as discussed by Novak and Lehky [18]. However, both in cases where optimization is applied to the identification problem and during any classic use of the optimization methods presented in [19], the decisive factor to support a successful optimization process consists in selecting the appropriate algorithm and correctly formulating the relevant objective function. The actual need of such a function becomes even more prominent in identification using optimization modules implemented within ANSYS Workbench [20], where the definition and computation of the objective function value have to be performed with an external program or script. The task embodying the inverse identification of unknown material parameters consists in utilizing the experimentally measured curves that characterize the relationship between the load L and the deformation d ( L-d curves). During the actual identification, this reference pattern is compared with the L-d curves produced by the nonlinear numerical simulations within the corresponding experiment. The basis of the comparison then rests in calculating the similarity ratio, which is represented by one or more numerical values and also prescribes the objective function. With respect to the formulation of the objective function, the optimization task is, in a given case, defined as the minimization of the similarity ratio. For the discussed purpose, it appears advantageous to employ the RMSE (Root-Mean-Square Error) ratio, an instrument that, according to [21], enables us to compare the differences between the values measured and those generated via a mathematical model; in this context, the authors of reference [22] analyze the application of the RMSE ratio within disciplines such as meteorology, economics, and demography. Considering the shape of the L-d curves, it is then possible, as shown in study [23], to exploit them in comparing the value of the surface below the loading curve with the maximum load value. Importantly, if the second one of the described variants is used, we also have to select a correct and robust algorithm to facilitate the optimization including multiple objective functions; this problem can be further encountered in the computation of more RMSE ratios for partial sections of the curve, whose positive impact is embodied in the analysis of the individual parameters‘ sensitivity to specific sections of the loading curve. With respect to the above-outlined conditions, the present article examines the effect exerted by different formulations of the objective function in a given identification task. For the identification proper, we chose the L-d curve measured during a three point bending test on a notched concrete beam, according to [24]. The numerical simulation of the experiment was performed with ANSYS via a nonlinear, multi-parametric material model of concrete adopted from the multiPlas library [25]. Generally, the paper aims to describe the applicability of the above-mentioned possibilities of formulating the objective function; the computation of the individual options was enabled by scripts created in Python. n order to analyze the suitability of the selected objective functions, we chose one L-d curve associated with the set of fracture tests published by Zimmermann et al. [24]. The specimen, a notched concrete beam manufactured from class C25/30 concrete and having the length l equal to 360 mm, height h of 120 mm, width w corresponding to 58 mm, and notch height of 40 mm, was configured for three point bending test; the vertical deformation d was measured in the middle of the span of 300 mm at the bottom side of the specimen. The cited article presented experimental and numerical research where the identification procedure based on utilization of neural network was used. The identification I I NPUT D ATA

8

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

process was aimed to finding values of basic mechanic-physical and fracture-mechanical properties of concrete specimens: modulus of elasticity E c , tensile strength f t and fracture energy G ft . The resultant statistical values of these properties were: modulus of elasticity E c = 38,9 GPa, tensile strength f t = 2,76 MPa and fracture energy G ft = 215,6 J/m 2 .

Figure 1 : The testing configuration.

G EOMETRY AND M ESH OF THE C OMPUTATIONAL M ODEL

T

he computational model and the nonlinear computation of the fracture experiment task were performed using ANSYS Workbench. To reduce the computing time, the geometry of the computational model was covered with a mesh of 2D finite elements (PLANE182) having the edge length of 6 mm, and the problem was solved as a plane stress task with the element thickness of w = 58 mm. The real support provided by steel bearings was, in the computational model, idealized via strain boundary conditions, where the vertical deformation was prevented at the location of the support, and the horizontal deformation was – with respect to the solvability of the task – restrained in the middle of the span at the upper side; such an arrangement then corresponded to the position of the introduced load. The notch was modeled in a simple manner, using a pair of parallel lines having a common node at the top of the notch.

Figure 2 : The computational model.

9

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

M ATERIAL M ODEL

T

o ensure both the nonlinear behavior and the identification of the corresponding material parameters, we selected a material model from the multiPlas database; this model is based on the plasticity surface derived by Willam and Warnke [5] and then modified by Menétrey [26], Menétrey and Warnke [13], and Klisinsky [27]. Further, the applied material model represents the evolutionary branch of nonlinear material models of concrete that exploit plasticity theory combined with instruments of nonlinear fracture mechanics to form a single model. This product then ranks among the group of models with non-associated plastic flow rule and is formulated such that it considers the invariants of the stress tensors and the deviatoric stress tensors; thus, the plasticity surface edges are softened, and the definition fidelity of the nonlinear behavior of concrete improves [10]. The formula for the yield surface as found in the programme manual [10] has the following form:

B ( , )  σ κ 2

  F A r e 2 ,     







  

( , ) 0 σ κ

(1)



 

MW

with the elliptic function r ( θ , e ) developed by Klisinsky [27] on the basis of Willam and Warnke´s findings [5] and where Ω( σ , κ ) is a hardening/softening function with a work-hardening law and A , B , C , D are model parameters containing basic mechanico-physical properties of concrete (uniaxial compressive strength f c , uniaxial tensile strength f t and biaxial compressive strength f b ) whose form is given by the following relationships:

f 1 1 1 6    f

 

f

f

b

t



 

(2)

A

  

c f 2



t

b

c f 2 3



B

(3)

2

C e 2 1   D e 2 1  

(4) (5)

with



    t t f f f f



2

2

2

2

t f   f   2



f f

f

 b c f f

b

c



e

(6)



2

2

2



f

2

b c

t

b

c

The Eq. (1) also contains coordinates of Haigh-Westergaard cylindrical space, where χ represents the height, ρ the radius and θ the azimuth. These coordinates are functions of the above mentioned invariants of stress tensor and can be written in the following manner: I 3 3   (7) J 2 2   (8) J J 3 3 2 3 3 cos3 2   (9)

The formula for the plastic potential takes the following form:

X Y 2  

  



Q

(10)

MW

10

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

where parameters X and Y are again related to uniaxial compressive strength f c

, uniaxial tensile strength f t

, biaxial

compressive strength f b

and the so-called dilatancy angle ψ .

c f X 2 tan 2 3(1 2 tan )      t f

(11)

t f X Y 2 2 3

 

(12)

and where

f

1

t

  

arctan

arctan

.

f

2

2

c

As regards using the finite element method, the model utilizes the concept of smeared cracks [28], and, thanks to exploiting Bazant’s crack band theory [29], it does not exhibit a negative dependence on the size of the finite element mesh. With respect to the above-described material model formulation consisting in a combination of several theoretical approaches, the total of 12 mechanico-physical and fracture-mechanical parameters were required to ensure the functionality of the model; a description of these parameters is presented in Tab. 1 [30].

Description

Parameter

Unit

E

[Pa]

Young’s modulus of elasticity

ν

[-]

Poisson’s ratio

f c

[Pa]

Uniaxial compression strength

f t

[Pa]

Uniaxial tension strength

K

[-]

Ratio between biaxial compressive strength and uniaxial compressive strength

ψ

Dilatancy angle (friction angle)

[ ͦ ]

ε ml

[-]

Plastic strain corresponding to the maximum load

G fc

[Nm/m 2 ] Specific fracture energy in compression

Ω ci

[-]

Relative stress level at the start of nonlinear hardening in compression

Ω cr

[-]

Residual relative stress level in compression

G ft

[Nm/m 2 ] Specific fracture energy in tension

Ω tr

[-]

Residual relative stress level in tension

Table 1 : The material model parameters.

I NVERSE I DENTIFICATION

he lack of knowledge of the input parameter values constitutes, together with the continuous theoretical development in the given field, a basic problem affecting the actual performance of advanced nonlinear material simulations of concrete structures. However, such knowledge deficiency can be advantageously eliminated via inverse analysis and experimental research. Yet the use of optimization techniques to identify the discussed parameters T

11

F. Hokes et alii, Frattura ed Integrità Strutturale, 39 (2017) 7-16; DOI: 10.3221/IGF-ESIS.39.02

still poses questions concerning the choice of a correct formulation for the relevant objective function. The inverse identification within this paper was carried out with an optimization module implemented in ANSYS Workbench; using such a computing system nevertheless required us to create external scripts in Python, and these were called during the batch calculation to compute the relevant objective function values. The whole process of inverse identification consisted in sensitivity analysis of the input material parameters to shape of the L-d curve and optimization itself which was used for minimization of the difference between the numerical and experimental L-d curves. The optimization algorithm was used for varying values of the material parameters and the parameters belonging to the curve with the lowest value of the objective function could be then considered as the sought material parameters. The calculation was performed automatically via ADPL macro that prepared geometry and mesh of the computational model, set up the material model with appropriate values of the material parameters, solved the task and called external Python script for calculation the of the objective function. Description of the Selected Objective Functions The basic objective function giving the difference between two curves was embodied in the RMSE ratio. The calculation of this function could not be performed directly, because the distribution of points on the reference and numerical curves was invariably different due to the varied runs of the solver. The mapping of the points on the numerical curve according to the reference curve was, within the script, resolved via linear interpolation. After aligning the points on the curves, we calculated the RMSE ratio according to the formula   i i y y n 2 * RMSE   (13) where y i * was the value of the force at the i -th point of the curve, and y i denoted the value of the force calculated using the nonlinear material model at the i -th point of the curve. Within the second optimization task, five optimization functions were created, formulated as the RMSE ratios calculated in five sectors evenly distributed along the curves. The prescription of these functions was identical with that shown in Eq. (1), the only difference being the number of points n , which corresponded to the number of points in the given sector. The third optimization task exploited two optimization functions. The former function was defined as the difference between the area ΔA Ld,ref under the reference L-d curve and the surface ΔA Ld,num below the numerically calculated L-d curve:

Ld A A A , ,    Ld ref

(14)

Ld num

the latter function, then, was defined similarly, as the difference between the maximum loading values L max,ref

and L max,num

in the form

 



L L max

L

(15)

ref

num

max,

max,

Sensitivity Analysis The actual identification of the material model parameters was invariably preceded by a sensitivity analysis aimed at mapping the space of the design variables and determining the sensitivity of the individual material parameters to the value of the objective function. For each identification task, we conducted 250 simulations, and uniform covering of the design space was ensured via the LHS method. The sensitivity of the individual material parameters to the output parameters was expressed using the Spearman correlation coefficient r s . The sensitivity analysis for the option with one RMSE optimization function showed that the highest sensitivity rate could be found in the elasticity modulus E , uniaxial tensile strength f t , and specific tensile fracture energy G ft . The high sensitivity rate of these parameters can be explained by the very basis of the examined problem: the simulated task is one with tensile bend. Very interesting results were obtained from the sensitivity analysis involving the subdivision of the material parameters into L-d curve sectors represented by 5 values of the RMSE ratio. In the given case, we identified that the first sector is

12