The Open Civil Engineering Journal

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1 Send Orders for Reprints to he Open Civil Engineering Journal he Open Civil Engineering Journal Content list available at: DOI: / RESEARCH ARICLE Anisotropi Nonlinear Elasti Model of Conrete and Seondar Development in ABAQUS Wei Sun and Yansheng Huang * Department of Civil Engineering South China Universit of ehnolog Guangzhou China Reeived: Marh Revised: Jul Aepted: August Abstrat: In order to solve the onvergene problem of onrete onstitutive in the softening phase an anisotropi nonlinear elasti onstitutive model () was proposed whih was developed based on the uniaxial onrete onstitutive relation in the Chinese ode for design of onrete strutures (GB ). he user material subroutine (UMA) based on is developed in ABAQUS software. he above UMA is applied to analze a simulation model in ABAQUS software. he result shows that ompared to the default plasti-damage onrete onstitutive in ABAQUS is an effetive and appropriate model to simulate the performane of onrete and it has improved the onvergene problem. Kewords: ABAQUS software Anisotropi nonlinear elasti onstitutive model Convergene problem Conrete onstitutive relation Simulation model User material subroutine. 1. INRODUCION Conrete is a widel used material in ivil engineering appliations. he orret understanding of onrete behavior is an essential part for design [1]. Man sholars have done a lot of experiments and theoretial studies and proposed a variet of onrete onstitutive models. hese models an be divided into 4 parts: 1. he linear elasti model 2. he nonlinear elasti model [2] 3. he plastiit theoretial model [3-7] 4. he other mehanis theoretial model [8]. Among these models the plastiit theoretial model has been used extensivel in reent ears to desribe the behavior of onrete. However still some disadvantages limit the development of this model suh as the parameters of plastiit model are diffiult to determine and the onvergene problem of onrete onstitutive in the softening phase. With the improvement of the material onstitutive theor more and more sholars onsidered the onrete material as anisotropi material. he anisotropi damage model an properl interpret the damage development in onrete material b desribing the underling miromehanial proesses of miro-defets [9 10]. In this paper an anisotropi nonlinear elasti onstitutive model () was proposed. is the simplified method based on the uniaxial onrete onstitutive relation in the Chinese ode for design of onrete strutures (GB ) and this model expands uniaxial onrete stress-strain behavior into three priniple diretions and simulates the anisotropi nonlinear behavior of onrete b using the seant moduli to alulate the updated stresses. he seant modulus in uniaxial onrete onstitutive relation is alwas positive whih an prevent the onvergene problem in the iterative proess of onrete onstitutive. Compared to the default plasti-damage onrete onstitutive in ABAQUS is an effetive and appropriate model to simulate the performane of onrete. 2. HE FORMULAS OF CONCREE CONSIUIVE RELAION he stress-strain behavior of the onrete material an be separated into tension and ompression omponents. he distintive behavior of onrete has inreased the omplexit of the onstitutive model. he tension part of the onrete stress-strain urve in the ode for design of onrete strutures (GB ) is desribed as Formula 1: * Address orrespondene to this author at the Department of Civil Engineering South China Universit of ehnolog Guangzhou China; el: ; vhuang@sut.edu.n / Bentham Open

2 616 he Open Civil Engineering Journal 2016 Volume 10 Sun and Huang (1 d ) E t (1) And the ompression part of the onrete stress-strain urve is desribed as Formula 2. (1 d ) E (2) where d t is a tensile damage parameter d is a ompressive damage parameter E is the initial elasti modulus of onrete. In the ode for design of onrete strutures (GB ) the formulas of d t and d are as follows: d t 5 t x x 1 [ ] 1 t 1 x t ( x1) x (3) ftr fr where t E E E r n E f tr r d n n1x 1 x 1 n 1 x 1 2 ( x 1) x r r. In Formula 3 x tr is the onrete peak tensile strain f tr is the orresponding value of onrete tensile strength at ε tr and α t is a desending parameter of tension part. In Formula 4 x r is the onrete peak ompressive strain f r is the orresponding value of onrete ompressive strength r at ε r and α is a desending parameter of ompression part. he orresponding values of α t and α ma refer to able 1. able 1. he values of α t and α orresponding to parameters of f tr and f r. tr (4) f tr (N/mm 2 ) α t f r (N/mm 2 ) α aking C30 onrete as an example the evolutions of d t and d with respet to variable x are shown in Figs. (1 and 2) respetivel. And the diagrammati skethes of the onrete uniaxial behavior under tension ondition and ompression ondition are shown in Figs. (3 and 4) respetivel. 1.0 d t C30 onrete x Fig. (1). Evolution of d t with respet to x

3 Anisotropi Nonlinear Elasti Model he Open Civil Engineering Journal 2016 Volume d C30 onrete Fig. (2). Evolution of d with respet to x x f tr + E 0 Random point E i 0 tr x + Fig. (3). Diagrammati sketh of onrete uniaxial behavior under tension ondition. f r Random point E 0 Fig. (4). Diagrammati sketh of onrete uniaxial behavior under ompression ondition. E i 0 r x 3. is proposed to simulate the onrete behavior subjeted to the omplex stress-strain ondition. he onrete onstitutive relation in is based on Formula1 and Formula 2 whih respetivel desribe the tension part and the ompression part of the onrete uniaxial behavior. How to appl the uniaxial onrete behavior into hree-dimensional spae is the ke point. In three-dimensional stress-strain onditions one of the main diffiulties in simulating the onrete behavior is the proper stiffness matrix in the iterative alulations. In this hapter the stiffness matrix used in will be desribed in detail. he stiffness matrix of onrete is denoted as [D]. In three-dimensional stress-strain onditions the stress-strain relation is desribed as Formula 5: [ D] where [ ] [ ].. x z x z xz x z x z xz From Formula 5 the stresses value an be alulated b the orresponding strains value. In order to expand the uniaxial onrete stress-strain relation into the three-dimensional onrete stress-strain relation there are 4 steps to ahieve this purpose. (5)

4 618 he Open Civil Engineering Journal 2016 Volume 10 Sun and Huang 3.1. Calulate Prinipal Strains from General Strains In the omplex stress-strain state the onrete behavior follows the uniaxial stress-strain relation in eah prinipal diretion. In general speaking the stresses values (or the strains values) in prinipal diretions are not the same as those in general diretions. he strains states in general diretions and in prinipal diretions are shown in Fig. (5). Z zx x z xz x z x z Y X General diretions Prinipal diretions Fig. (5). he strains states in general diretions and in prinipal diretions. he transformation from the general strains state [ x z x z xz ] to the prinipal strains state [ ] an be aomplished b the prinipal strain onversion matrix [ 1 ]. he formula is expressed as follows [ 000] [ ][ ] x z x z xz (6) It is important to notie that ε x =γ x /2 ε z =γ z /2 ε xz =γ xz /2. he prinipal strain onversion matrix [ 1 ] is expressed in Formula 7 [11]: l1 m1 n1 2l1m 1 2m1n 1 2n1l l2 m2 n2 2l2m2 2m2n2 2n2l l3 m3 n3 2l3m3 2m3n3 2n3l [ 3 1] l1l 2 m1m 2 n1n 2 l1m2 l2m1 m1n 2 m2n1 n1l 2 n2l 1 l2l3 m2m3 n2n3 l2m3 l3m2 m2n3 m3n2 n2l3 n3l2 l3l1 m3m1 n3n1 l3m1 l1m3 m3n1 m1n 3 n3l1 n1l 3 (7) where l 2 +m 2 +n 2 =1 and the values of l m and n an be alulated b Formula 8. x z xz ( i ) li [ ( )] [ ( )] [( )( ) ] xz x z ( x i ) mi li x z xz ( i ) 2 ( x i )( i ) x ni li x z xz ( i ) i x z xz i x xz z x i x i i x 3.2. Calulate Effetive Prinipal Strains from Prinipal Strains In the effetive prinipal strains [ε 1 * ε 2 * ε 3 * 0 0 0] are exatl following the uniaxial stress-strain rules whih an be alulated from the prinipal strains [ε 1 ε 2 ε ] b Formula 9. (8)

5 Anisotropi Nonlinear Elasti Model he Open Civil Engineering Journal 2016 Volume * * * [ 000] [ ][ 000] (9) where [ 2 ] is the effetive prinipal strain onversion matrix and it is desribed as follow: 2 (1 ) (1 )(1 2 ) (1 )(1 2 ) (1 )(1 2 ) (1 ) (1 )(1 2 ) (1 )(1 2 ) (1 )(1 2 ) (1 ) (1 )(1 2 ) (1 )(1 2 ) (1 )(1 2 ) (10) where υ is Poisson s ratio of the onrete material Calulate Prinipal Stresses from Effetive Prinipal Strains In eah prinipal diretion the relation between prinipal stresses and effetive prinipal strains follows the hange law of uniaxial onrete onstitutive relation and the alulation of the seant modulus in eah priniple diretion is independent of eah other. he relation between prinipal stresses and effetive prinipal strains is desribed as Formula 11: * * * D [ 000] [ ][ 000] (11) where [D 1 ] is the seant matrix and it is desribed as Formula 12: where E i is the seant modulus whih is orresponding to the effetive prinipal strains state. he formula of E i is as follows: In the tension ondition E i =(1-d ti )E. In the ompression ondition E i =(1-d i )E Calulate General Stresses from Prinipal Stresses D 1 E E E he general stresses is orresponding to the global oordinate sstem. In the iterative proess it is onvenient to update the stresses in the global oordinate sstem. he general stresses an be alulated from the prinipal stresses b Formula 13: (12) 1 x z x z xz [ ] [ ] [ 000] (13) where [ 1 ] is the prinipal strain onversion matrix. It an be onluded from the Formulas of and 13 that the expression of the onrete stiffness matrix an

6 620 he Open Civil Engineering Journal 2016 Volume 10 Sun and Huang be written in the following form. 1 D= D (14) herefore the general stresses an be alulated from the general strains b Formula 15. x z x z xz D x z x z xz [ ] [ ][ ] (15) Analogous to Formula 5 Formula 15 is the stress-strain relationship used in. 4. HE SECONDARY DEVELOPMEN IN ABAQUS How to appl in ABAQUS software? First it is needed to know the iteration prinipal in ABAQUS [12-15]. he main purposes of the onstitutive iteration in ABAQUS are the stiffness matrix alulation and the stresses update. he stiffness matrix [D] an be alulated from Formula 14 and the stresses σ ij n+1 are updated based on the parameters of σ ij n ε ij n and dε ij n. he updated stresses σ ij n+1 are alulated b Formula 16: n 1 = D ( n d n ) ij ij ij (16) n1 n1 n1 n1 n1 n1 n1 n n n n n n n where ij =[ x z x z xz ] ij =[ x z x z xz ] n n n n n n. n. d =[ d d d d d d ]. It should be notied that eah of the seant modulus E 1 E 2 and E 3 in the stiffness matrix [D] has two forms whih respetivel orresponding to the tension state and the ompression state. he effetive prinipal strains [ε 1 * ε 2 * ε 3 * 0 0 0] are the ke fators to judge the states. As the effetive prinipal strain ε i * is positive in the orresponding prinipal diretion Conrete material is onsidered to be in a state of tension and d ti should be seleted as a damage parameter in the alulation proess of E i. In ontrast as the effetive prinipal strain ε i * is negative in the orresponding prinipal diretion Conrete material is onsidered to be in a state of ompression and d i should be seleted as a damage parameter in the alulation proess of E i. From Formula 16 it an be known that the updated stresses are alulated diretl from the updated strains and it is not affeted b the last step stresses. he iteration proess in Formula 16 is alled full inremental iterative method. In this method the seant moduli in the stiffness matrix [D] are alwas kept positive. hat means there is no singular problem in the stiffness matrix during the onstitutive iteration [16]. In other words in the proess of solving displaement it an diretl alulate the inverse stiffness matrix [D] -1 and there is no onvergene problem during the onstitutive iteration. he algorithm flow hart of the onstitutive iteration in ABAQUS is as Fig (6): Enter the neessar material properties parameters of E ν α t α ε tr ε r f tr and f r. Extrat strain and strain inrement in urrent load step. Calulate the iteration stiffness matrix [D]. Update stress and strain. No Loading omplete? End Yes Fig. (6). Flow hart of onstitutive iteration in ABAQUS.

7 Anisotropi Nonlinear Elasti Model he Open Civil Engineering Journal 2016 Volume HE SIMULAION EXAMPLE In order to text the proposed a simulation example is taken in ABAQUS software whih is shown in Fig. (7). Beause the main purpose of the simulation is to verif the orretness of the algorithm in the finite element model size is hosen as 1m 1m 1m and the mesh size is hosen as 1m. Meanwhile the parameters of the onrete material propert applied in the example are shown in able 2. Uniaxial tensile loading ensile loading with onstraint Shear loading onl Uniaxial ompressive loading Compressive loading with onfining pressure Shear loading and ompressive loading Fig. (7). he simulation model in ABAQUS. able 2. he parameters of the onrete material propert. Parameters E ν α t α f tr f r ε tr ε r Value 30000MPa MPa 30MPa and the plasti-damage model are taken to analze the simulation model respetivel. he alulated stress-stain urves in various loading onditions are shown in Figs. (8-13) MPa Plasti-damage model Code urve Fig. (8). Stress-strain urves under uniaxial tensile loading MPa Plasti-damage model Code urve Fig. (9). Stress-strain urves under uniaxial ompressive loading.

8 622 he Open Civil Engineering Journal 2016 Volume 10 Sun and Huang MPa Plasti-damage model Code urve Fig. (10). Stress-strain urves under tensile loading with onstraint. In Figs. (8-11) it an be seen that under the uniaxial tensile loading (or the uniaxial ompressive loading) without onfining pressure there is a generall good agreement in the shape of the urves alulated from and the plasti-damage model. But when the onfining pressure is under onsideration the urves alulated from and the plasti-damage model have a bit differene. Compared to the plasti-damage model urve the peak strain in urve is redued due to the influene of the surrounding pressure and beause the plasti damage-model an not effetivel simulate the mehanial behavior of onrete under high onfining pressure (whih is mentioned in Abaqus Analsis User s Guide) the shape of the urve alulated b the plasti-damage model in Fig. (11) is obviousl different from others. In Figs. (12 and 13) it is shown that the urves alulated b and the plasti-damage model have presented the same tenden while the simulation model is under shear loading and under ompliated loading (shear loading and ompressive loading) MPa Plasti-damage model Code urve Fig. (11). Stress-strain urves under ompressive loading with onfining pressure τ MPa Plasti-damage model γ Fig. (12). Stress-strain urves under shear loading. It should be notied that the stress-strain relation in prinipal diretions are stritl following the ode urve when

9 Anisotropi Nonlinear Elasti Model he Open Civil Engineering Journal 2016 Volume is taken to analze whih an be proved in Figs. (14 and 15). It means in three-dimensional spae suessfull represented the ode urves in prinipal diretions. As we have seen whether an effetivel simulate the onrete performane is depended on the seleted uniaxial onstitutive model. Meanwhile beause the hosen onstitutive model is not ontained the stiffness reover fators is not suitable to simulate the performane of onrete under li loading τ MPa Plasti-damage model γ 10-2 Fig. (13). Stress-strain urves under shear and ompressive loading σ MPa Code urve ε 10-4 Fig. (14). ensile stress-strain urves in prinipal diretion σ MPa Code urve ε 10-3 Fig. (15). Compressive stress-strain urves in prinipal diretion. CONCLUSION In this paper was proposed to simulate the onrete behavior and a simulation model was taken to analze. he result shows that ompared to the default plasti-damage onrete onstitutive in ABAQUS is an effetive and appropriate model to simulate the performane of onrete and it has improved the onvergene

10 624 he Open Civil Engineering Journal 2016 Volume 10 Sun and Huang problem. However still has the following limitations. 1) he alulation aura of is depended on the seleted onstitutive model. 2) is not suitable to simulate the performane of onrete under li loading. For improving the above defiienies further work is being studied. CONFLIC OF INERES he authors onfirm that this artile ontent has no onflit of interest. ACKNOWLEDGEMENS he projet was supported b the National Nature Siene Foundation of China (Grant No ). REFERENCES [1] G.Z. Voiadjis Z.N. aqieddin and P.I. Kattan "Anisotropi damage-plastiit model for onrete" Int. J. Plast. vol. 24 pp [ [2] Y.R. Kim D.H. Allen and D.N. Little "Computational onstitutive model for prediting nonlinear visoelasti damage and frature failure of asphalt onrete mixtures" Int. J. Geomeh. vol. 7 pp [ [3] S. Ananiev and J. Ožbolt Plasti-Damage Model for Conrete in Prinipal Diretions. Eprint Arxiv [4] H. Hu and W.C. Shnobrih "Constitutive modeling of onrete b using nonassoiated plastiit" J. Mater. Civ. Eng. vol. 1 pp [ [5] L. Jason A. Huerta and G. Pijaudier-Cabot "An elasti plasti damage formulation for onrete: appliation to elementar tests and omparison with an isotropi damage model" Comput. Methods Appl. Meh. Eng. vol. 195 pp [ [6] K. Maekawa and J. akemura "Plastiit in onrete nonlinearit under triaxial onfinement" J. Mat. Conr. Strut. Pave. vol. 18 pp [7] A.K. Li X.L. Du D.C. Lu and S.. Cao "A three- dimensional elasti-plasti onstitutive model for onrete" China Civil Eng. J. vol. 45 pp [in Chinese]. [8] S. Ramtani Y. Berthaud and J. Mazars "Orthotropi behavior of onrete with diretional aspets: modelling and experiments" Nul. Eng. Des. vol. 133 pp [ [9] S. Fihant C.L. Borderie and G. Pijaudier-Cabot "Isotropi and anisotropi desriptions of damage in onrete strutures" Meh. Cohes.Frit. Mater. vol. 4 pp [ [10] J.W. Ju "Isotropi and anisotropi damage variables in ontinuum damage mehanis" J. Eng. Meh. vol. 116 pp [ [11]. Wang "he disussion on analti expressions of primar stress and diretion in stressed spae" Indust. Constr. vol. 36 pp [in Chinese]. [12] J.G. Nie and Y.H. Wang "Comparison stud of onstitutive model of onrete in ABAQUS for stati analsis of strutures" Eng. Meh. vol. 30 pp [in Chinese]. [13] W.Z. Dong "hree-dimensional nonlinear finite element analsis of reinfored onrete strutures" J. Shenang Arhit. Civil Eng Inst. vol. 7 pp [in Chinese]. [14] H. Qi Y.G. Li X.M. Chen J. Duan and J.Y. Sun "Stud on uniaxial onstitutive model of onrete in ode for design of onrete strutures and development in ABAQUS" Build. Strut. vol. 44 pp [in Chinese]. [15] M. Liu Q. Zhang and Z.Q. Liu "Seondar development of elasti-brittle-plasti onstitutive model and its appliation based on ABAQUS" HongShui River vol. 29 pp [in Chinese]. [16] B.F. Zhu he Finite Element Method heor and Appliations. Water Resoures and Eletri Power Press: Beijing 2009 pp Sun and Huang; Liensee Bentham Open his is an open aess artile liensed under the terms of the Creative Commons Attribution-Non-Commerial 4.0 International Publi Liense (CC BY-NC 4.0) ( whih permits unrestrited non-ommerial use distribution and reprodution in an medium provided the work is properl ited.