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Investigation for plastic damage constitutive models of the concrete material
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ScienceDirect
Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000
www.elsevier.com/locate/procedia
ScienceDirect Procedia Engineering 210 (2017) 71–78
6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading, PROTECT2017, 11-12 December 2017, Guangzhou (Canton), China
Investigation for plastic damage constitutive models of the concrete material Wei Demin , He Fukang一 一
School of Civil and Transportation, South China University of Technology, State Key Laboratory of Subtropical Building Science, Guangzhou 510640, Guangdong, China
Abstract In this paper, some simple stress-strain relationships of the concrete material recommended in relevant Codes are appropriately simplified, then the damage factors of the simplified plastic damage constitutive model is determined based on Sidiroff’s energy equivalence principle. Mechanical characteristics of the concrete material under the simple tension or compression are analyzed by Finite element method. Through the comparison of numerical analysis results and Code constitutive relations, the damage factors of the simplified plastic damage constitutive model is verified. The unreinforced concrete beam static tests by Petersson is simulation analyzed by the nonlinear finite element method and plastic damage constitutive model. The effect of the unit size and the different linear softening constitutive relation on the analysis results are considered. The results show that there is no obvious size effect on the plastic damage analysis results based on fracture cracking criterion, the results of the bilinear softening constitutive analysis have good accuracy, and the form of softening constitutive relation has a great influence on the result. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Strengthening of Structures Structures under under Extreme Extreme Loading. Loading Keywords: finite element method;concrete;plastic damage;nonlinear analysis;
1. Introduction The stress-strain relationship of concrete under multiaxial stress plays an important role in the finite element analysis of the nonlinear response of concrete structures. There are a lot of constitutive model of concrete based on
* Corresponding author. Tel.: 15521132772 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading. 10.1016/j.proeng.2017.11.050
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
72 2
the experimental and theoretical study, such as linear elastic model, nonlinear elastic model, plastic theory model, viscous - plastic theory, internal theory, fracture theory , damage theory and so on. However, due to the complex nature of concrete materials, there is no common constitutive model of concrete. Plastic Damage Constitutive Model it is assumed that the damage of concrete material is mainly caused by tensile cracking and compression crushing. Some studies have been made on the damage factors and related parameters in the plastic constitutive model of concrete, Yao Guohuang etc. [1] studied the parameters in the plastic damage model and applied it to the static analysis of steel-concrete composite structures. Lei Tuo etc. [2] through the analysis of reinforced concrete simply supported beam, discussed the dilation angle, viscosity parameter, tensile stiffening and other constitutive parameters on the analysis results. Cao Ming [3] proposed a plastic damage factors calculation method, and analysed the reinforced concrete simply supported beam. Zhang Jin etc. [4] carried out the relevant damage parameters for the concrete uniaxial constitutive relation given by the relevant norm. Qin Hao etc. [5] studied the method of taking the concrete damage factors in ABAQUS program. In this paper, the calculation method of plastic damage parameters in ABAQUS program is further studied, and the modified constitutive model of concrete plastic damage is proposed and verified. The nonlinear static plastic damage analysis of the plain concrete beam is carried out, and the results are compared with the experimental results. 2. Plastic damage model The plastic damage constitutive model in ABAQUS program is suitable for concrete and rock [6,7] quasi-brittle materials, which can simulate the tensile cracking and compressive crushing of concrete materials, considering the isotropic elastic damage and plastic behavior of materials. In general, the strain rate ε can be decomposed into elastic strain rate ε el and plastic strain rate ε pl :
ε = ε el + ε pl
\* MERGEFORMAT (1)
And the elastic damage constitutive relationship is: σ (1 d )Del0 : (ε ε pl ) Del : (ε ε pl )
\* MERGEFORMAT (2)
where Del0 and Del are the initial (undamaged) elastic stiffness of the concrete and the degraded elastic stiffness ,and d is the damage factor, the scalar stiffness degradation variable, which can take values in the range of zero (undamaged material) to one (fully damaged material). The damage factors d c and d t represent the stiffness degradation rate of the concrete caused by the damage of the concrete during compression and tension,under the condition of uniaxial stress,from the formula (2) ,get the tensile and compressive damage constitutive relationship, as shown in Fig. 1.
(a)
(b)
Fig. 1. Uniaxial damage constitutive curve of concrete:(a) uniaxial tension; (b) uniaxial compression.
The yield function F which consider the yield strength of tension and compression is [6,7]:
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
F (σ , ε pl )
1 (q 3 p (ε pl ) ˆ max (ˆ max )) c (c pl ) 0 1
73 3
\* MERGEFORMAT (3)
1 where and both are dimensionless material constants, p σ ：I is the effective hydrostatic pressure, σ is the 3
3 S : S is the Mises equivalent effective stress, 2 is the deviatoric part of the effective stress tensor σ , ˆ max is the algebraically maximum
effective stress tensor, ε pl is the equivalent plastic strain tensor, q where S pI σ
eigenvalue of σ ,the function ( ε pl ) is given as (ε pl )
c (cpl ) (1 ) (1 ) , where t and c are the effective t (tpl )
tensile and compressive cohesion stresses, respectively, the coefficient can be determined from the initial c0 equibiaxial and uniaxial compressive yield stress, b0 and c0 ,as b 0 . 2 b 0 c 0 The relationship between the plastic strain rate ε pl and the plastic potential function G (σ ) is: ε pl
G (σ ) σ
\* MERGEFORMAT (4)
where, 0 is a plastic multiplier. For the non-associated plastic flow rule, the plastic potential function is the Drucker-Prager hyperbolic function: 2 G(σ ) （e t0 tan） q 2 ptan
\* MERGEFORMAT (5)
where is the dilation angle, t0 is the uniaxial tensile stress at failure, e is eccentricity parameter, the flow potential tends to a straight line as the eccentricity tends to zero. 3. Damage factor determination method When the ABAQUS plastic damage constitutive model is used for analysis, the compression, tensile stress and the corresponding non-elastic strain and damage factor of the material are required. In this paper, the damage factors are calculated based on the Sidiroff’s energy equivalent principle [8]. The Sidiroff's energy equivalent principle assumes that the stress acting on the damaged material and the nondamaged material produces the same energy in the form of the same, the elastic residual energy of the non-damaged material and the equivalent elastic energy of the damaged material are: W0 e 2 （ / 2 E0）
\* MERGEFORMAT (6)
Wd e 2 （ / 2 E0） 2 （ / 2 Ed）
\* MERGEFORMAT (7)
. 1 d From formula (7),the equivalent elastic modulus is derived as Ed E0 (1 d ) 2 and the uniaxial constitutive relation and damage factor of the damaged material can be obtained: where, the effective stress is
E0 (1 d ) 2
\* MERGEFORMAT (8)
Wei Demin al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/etProcedia Engineering 00 (2017) 000–000
74 4
d 1
\* MERGEFORMAT (9)
E0
4. Verification of damage factors Elastic modulus is the main indicator of material deformation performance, determined by the relevant test [9].When the concrete material is subjected to uniaxial compression and the stress is greater than 0.3~0.5 times its compressive strength, the microfracture develops into the unstable stage, and the material exhibits obvious nonlinearity. The material is in the linear elastic phase under the action of uniaxial tension less than 0.6~0.8 times its ultimate tensile strength. Therefore, the values of the modulus of elasticity under uniaxial compression take the secant modulus at half of the working stress, and the values of the uniaxial tensile modulus of elasticity generally take the secant modulus at half of the work stress, both are equal as E0 [9]. The uniaxial tension constitutive relationship given in the norm [10] is simplified as: E x / f t y 0 t 1.7 x / [ t ( x 1) x]
( x 1)
\* MERGEFORMAT (10)
( x 1)
where y / f t , x / t , f t is the representative value of the uniaxial tensile strength of the concrete, and its value can be taken according to the actual structural analysis, respectively,the characteristic value f tk , the design value f t or the mean value f tm , t is the corresponding peak tensile strain corresponding to f t , t is the stress-strain curve descending segment parameters. The ascending and descending segments of the constitutive relation of concrete under uniaxial compression given in the norm [10] are simplified as: E0 c x / f c ( 0.5 f c ) y 2 3 a x (3 - 2 a ) x ( a 2) x (0.5 f c f c )
\* MERGEFORMAT (11)
y x / [ d ( x 1) 2 x]
\* MERGEFORMAT (12)
( x 1)
where y / f c , x / c , f c is the representative value of the uniaxial compressive strength of the concrete, and its value can be taken according to the actual structural analysis, taking the characteristic value f ck , design value f c or mean value f cm ,respectively, c is the peak pressure strain corresponding to f c , a and d are the parameters of the stress-strain curve rising and falling segments, respectively. Taking C40 grade concrete as an example to calculate the damage factor. The material properties are: uniaxial compressive strength characteristic value f ck =26.8N/mm 2 , uniaxial tensile strength characteristic value f tk =26.8N/mm 2 , elastic modulus E0 =2.7393 10 4 N/mm 2 , Poisson's ratio =0.2 , dilation angle =30 0 , eccentricity e 0.1 , the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress is f b 0 / f c 0 1.16 , the ratio of the second stress invariant on the tensile meridian to that on the compressive
meridian K c 0.6667 , viscosity parameter 0.0001 . Table 1 is the concrete damage factors calculated from Eq. (11), (13) to (15). Table 1. C40 grade concrete damage factors. Yield stress in compression ( MPa )
Inelastic Strain (10 3 )
Compressive damage variable
Stress after cracking ( MPa )
Cracking Strain (10 3 )
Tensile damage variable
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
75 5
13.941
0
0.000
2.390
0
0.000
24.505
0.218
0.103
2.180
0.025
0.128
26.800
0.611
0.215
1.885
0.053
0.250
24.792
1.161
0.338
1.750
0.067
0.301
21.214
1.769
0.448
1.425
0.105
0.425
18.926
2.17
0.509
1.196
0.14
0.512
15.176
2.943
0.602
0.947
0.192
0.610
10.909
4.211
0.706
0.719
0.27
0.702
6.956
6.421
0.805
0.485
0.436
0.803
5.159
8.394
0.852
0.376
0.597
0.850
3.377
12.274
0.900
0.262
0.95
0.900
3.025
13.558
0.910
0.239
1.073
0.910
2.645
15.32
0.921
0.216
1.231
0.920
2.303
17.399
0.931
0.197
1.389
0.928
1.969
20.113
0.940
1.633
23.94
0.950
1.535
25.374
0.953
Taking the standard specimen of concrete prism as an example, the element type is three-dimensional solid element C3D8R, the specimen which is divided into 54 units and the unit size is 150mm 150mm 300mm . Through the uniaxial displacement loading, the reaction force and displacement are obtained, and the relationship between the uniaxial stress and strain is calculated. The comparison between the calculated results and the norm is given in Fig.2:
(a)
(b)
Fig. 2. Uniaxial stress-strain curve: (a) Uniaxial compression; (b) Uniaxial tension.
It can be seen from Fig. 2 that the peak stress and peak strain of the concrete uniaxial constitutive curves obtained by finite element simulation are in good agreement with the code results, the peak stress and peak strain of uniaxial compression were 27.33MPa and 1.72E-3,which were 1.98% and 8.17% higher than the code value respectively. The uniaxial tensile stress and peak strain are 2.32MPa and 9.42E-5, respectively, which are 3.03% smaller than the Code value and 8.03% higher than the Code value. The calculated results show that the deviation of the stress results is smaller, and the calculated peak strain is larger than the code value because the taking value of the elastic modulus in the paper is smaller than the Code. And the other grades of concrete can be verified in the same way.
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
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5. Test simulation Petersson [11] studied the plain concrete beam shown in Fig. 3, the test results show that the nonlinear behavior of the simply supported beam caused by type I tensile cracks. The geometric dimensions are: l 2.0m , b 0.05m , h 0.2m , d 0.04m , a 0.1m .
(a)
(b)
Fig. 3. Geometric dimensions and finite element model of notch beam: (a) Geometric dimensions; (b) Finite element meshing
The simulation was carried out using the plastic damage model described above. Taking the semi-structure for analysis according to the symmetry.Using the four-node plane stress reduction integral unit CPS4R,in order to study the influence of mesh size on the analysis results, the coarse, medium and fine meshing units are used, and the total number of semi-structural elements is 352,1120,4480,respectively. The beam is loaded by prescribing the vertical displacement at the center of the beam until it reaches a value of 0.0015m and the Riks method is used. The material properties of concrete are: elastic modulus E0 30 Gpa , Poisson's ratio 0.2 , density 0 2400 kg/m3 , uniaxial tensile strength f t 3.33 MPa , fracture energy G If 124 N/m . In this paper, the tensile stiffening parameters [12] are using the stress-cracking displacement relationship based on the fracture energy cracking criterion. The linear tenion[13]softening curve, the European Model Code [14], Petersson [11] proposed bilinear tension softening curve are using for analysis, respectively, as shown in Fig. 4. The tensile damage of concrete is considered and assuming that the tensile damage factor increases linearly with the crack displacement, it reaches the maximum value until the ultimate crack displacement,is shown in Fig. 5.
Fig. 4. Softening constitutive relationship.
Fig. 5. Tensile damage factor.
Fig. 6 and Fig. 7 show the comparison of the results of finite element model analysis in different meshes. It can be seen from the figure that the results of coarse, medium and fine mesh are consistent, so the stress-cracking displacement based on fracture energy cracking criterion there is no obvious size sensitivity in response.
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
Fig. 6. Result of linear softening constitutive model.
77 7
Fig. 7. Result of MC bilinear softening constitutive model.
Fig. 8 shows the comparison of the results of the different tension softening constitutive models under mediummeshing and the results of the test.
Fig. 8. Comparison of simulated results of Petersson notch beam.
In the test [11], the fracture energy parameters of concrete are 115 ~ 137 N / m, the average value is 124 N / m, and the corresponding test peak load are 727.80N and 800N respectively, and the average value is 763.9N, the peak displacements are 0.334 mm and 0.396 mm, respectively, and the average value is 0.365 mm. The peak load calculated by the linear softening constitutive model, the European norm and the Petersson proposed bilinear softening constitutive model are 894.78N, 805.06N and 752.00N, respectively,the corresponding calculated displacements are 0.565mm, 0.472mm, 0.410mm. The results of single linear softening constitutive model,is too stiff compared with the experimental observations of Petersson . The calculated peak load is 17.1% higher than that of the test, and the peak displacement is 54.8% larger than that of the test,the analysis results are less accurate .The peak load calculated by the bilinear softening constitutive model proposed by the European and Petersson is 5.4% larger than the test and 1.6% smaller than the test respectively,the deviation of the load is smaller and the deviation of the displacement is larger.The results show that the bilinear softening constitutive model has better accuracy than the single linear softening constitutive model. The difference of the bilinear softening form also affects the analysis results. Fig. 9 is the contour of PEEQT, PE Max Principal, S Max Principal ,DAMAGET of the FEM ananlysis of Petersson proposed bilinear softening constitutive model in medium mesh, which are referred to as the tensile equivalent plastic strains, the maximum principal plastic strain, tensile damage ,maximum principal stress. We can assume that cracking initiates at points where the tensile equivalent plastic strain is greater than zero and the maximum principal plastic strain is positive.
78 8
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
(a)
(b)
(c)
(d)
Fig. 9. Contour of FEM analysis: (a)PEEQT; (b) PE Max Principal; (c) DAMAGET ; (d) S Max Principal.
6. Conclusion In this paper, two kinds of tensile stiffening methods of ABAQUS plastic damage model are discussed by finite element analysis of uniaxial tension and compression of concrete and the experiment of plain concrete notch beam. Describing the strain softening characteristic of the cracking concrete based on the relationship between the subsequent failure stress and the cracking strain through the normative uniaxial constitutive relation and the damage factor determination method in this paper is available. The relevant factors of the tensile stiffening parameters and how to select the reasonable tensile stiffening parameters need further study. In terms of using the stress-cracking displacement relation based on the fracture energy cracking criterion to define the strain softening characteristics of the cracking concrete, the analysis results show that there is no obvious element size effect. The results of the bilinear softening constitutive model have better accuracy, also the influence of different softening constitutive curves on the analysis results needs further study. References [1] Yao G H, Huang Y J, Song B D, et al. Analysis for Static Behaviour of Steel and Concrete Composite Members with Plastic-Damage Model[J]. Progress in Steel Building Structures. 2009. [2] Lei T. Application of Damaged Plasticity Model for Concrete[J]. Structural Engineers. 2008. [3] G.R. Mettam, L.B. Adams, How to prepare an electronic version of your article, in: B.S. Jones, R.Z. Smith (Eds.), Introduction to the Electronic Age, E-Publishing Inc., New York, 1999, pp. 281–304. [3] Cao M. Research on Damage Plastic Calculation Method of ABAQUS Concrete Damaged Plasticity Model[J]. Transportation Standardization. 2012. [4] Zhang J, Wang Q, Hu S, et al. Parameters Verification of Concrete Damaged Plastic Model of ABAQUS[J]. Building Structure. 2008, 38(8): 127-130. [5] Qin H, Zhao X. Study on the ABAQUS Damage Parameter in the Concrete Damage Plasticity Model[J]. Structural Engineers. 2013. [6] Lubliner J, Oliver J, Oller S, et al. A plastic-damage model for concrete[J]. International Journal of Solids & Structures. 1989, 25(3): 299-326. [7] Lee J H, Fenves G L. Plastic-damage model for cyclic loading of concrete structures[J]. JOURNAL OF ENGINEERING MECHANICSASCE. 1998, 124(8): 892-900. [8] LI Zhaoxia. Damage mechanics and its applications[M]. Beijing: Science Press,2002:7-30. [9] Guo Z H.Theory of reinforced concrete[M]. Beijing:Tsinghua University Press.2013. [10] Ministry of Construction of the People’s Republic of China.GB50010-2002 Code for design of concrete structures[M]. Beijing:China Architecture & Building Press.2002. [11] Petersson P E. Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials[J]. Report No.TVBM1006,Division of Building Materials,University of Lund,Sweden. [12] Wang J C,Chen Y K. Application of ABAQUS in Civil Engineering[M].Hangzhou: Zhejiang University Press.2006. [13] Wang Q Y. Research on concrete softening relationship from notched three-point bending beam based on evolutionary algorithms[D]. Hangzhou: Zhejiang University of Technology.2012. [14] Betonbau. fib Model Code for Concrete Structures 2010[J]. Ernst & Sohn. 2013.
ScienceDirect
Available online at www.sciencedirect.com Procedia Engineering 00 (2017) 000–000
www.elsevier.com/locate/procedia
ScienceDirect Procedia Engineering 210 (2017) 71–78
6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading, PROTECT2017, 11-12 December 2017, Guangzhou (Canton), China
Investigation for plastic damage constitutive models of the concrete material Wei Demin , He Fukang一 一
School of Civil and Transportation, South China University of Technology, State Key Laboratory of Subtropical Building Science, Guangzhou 510640, Guangdong, China
Abstract In this paper, some simple stress-strain relationships of the concrete material recommended in relevant Codes are appropriately simplified, then the damage factors of the simplified plastic damage constitutive model is determined based on Sidiroff’s energy equivalence principle. Mechanical characteristics of the concrete material under the simple tension or compression are analyzed by Finite element method. Through the comparison of numerical analysis results and Code constitutive relations, the damage factors of the simplified plastic damage constitutive model is verified. The unreinforced concrete beam static tests by Petersson is simulation analyzed by the nonlinear finite element method and plastic damage constitutive model. The effect of the unit size and the different linear softening constitutive relation on the analysis results are considered. The results show that there is no obvious size effect on the plastic damage analysis results based on fracture cracking criterion, the results of the bilinear softening constitutive analysis have good accuracy, and the form of softening constitutive relation has a great influence on the result. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Strengthening of Structures Structures under under Extreme Extreme Loading. Loading Keywords: finite element method;concrete;plastic damage;nonlinear analysis;
1. Introduction The stress-strain relationship of concrete under multiaxial stress plays an important role in the finite element analysis of the nonlinear response of concrete structures. There are a lot of constitutive model of concrete based on
* Corresponding author. Tel.: 15521132772 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading.
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of the 6th International Workshop on Performance, Protection & Strengthening of Structures under Extreme Loading. 10.1016/j.proeng.2017.11.050
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
72 2
the experimental and theoretical study, such as linear elastic model, nonlinear elastic model, plastic theory model, viscous - plastic theory, internal theory, fracture theory , damage theory and so on. However, due to the complex nature of concrete materials, there is no common constitutive model of concrete. Plastic Damage Constitutive Model it is assumed that the damage of concrete material is mainly caused by tensile cracking and compression crushing. Some studies have been made on the damage factors and related parameters in the plastic constitutive model of concrete, Yao Guohuang etc. [1] studied the parameters in the plastic damage model and applied it to the static analysis of steel-concrete composite structures. Lei Tuo etc. [2] through the analysis of reinforced concrete simply supported beam, discussed the dilation angle, viscosity parameter, tensile stiffening and other constitutive parameters on the analysis results. Cao Ming [3] proposed a plastic damage factors calculation method, and analysed the reinforced concrete simply supported beam. Zhang Jin etc. [4] carried out the relevant damage parameters for the concrete uniaxial constitutive relation given by the relevant norm. Qin Hao etc. [5] studied the method of taking the concrete damage factors in ABAQUS program. In this paper, the calculation method of plastic damage parameters in ABAQUS program is further studied, and the modified constitutive model of concrete plastic damage is proposed and verified. The nonlinear static plastic damage analysis of the plain concrete beam is carried out, and the results are compared with the experimental results. 2. Plastic damage model The plastic damage constitutive model in ABAQUS program is suitable for concrete and rock [6,7] quasi-brittle materials, which can simulate the tensile cracking and compressive crushing of concrete materials, considering the isotropic elastic damage and plastic behavior of materials. In general, the strain rate ε can be decomposed into elastic strain rate ε el and plastic strain rate ε pl :
ε = ε el + ε pl
\* MERGEFORMAT (1)
And the elastic damage constitutive relationship is: σ (1 d )Del0 : (ε ε pl ) Del : (ε ε pl )
\* MERGEFORMAT (2)
where Del0 and Del are the initial (undamaged) elastic stiffness of the concrete and the degraded elastic stiffness ,and d is the damage factor, the scalar stiffness degradation variable, which can take values in the range of zero (undamaged material) to one (fully damaged material). The damage factors d c and d t represent the stiffness degradation rate of the concrete caused by the damage of the concrete during compression and tension,under the condition of uniaxial stress,from the formula (2) ,get the tensile and compressive damage constitutive relationship, as shown in Fig. 1.
(a)
(b)
Fig. 1. Uniaxial damage constitutive curve of concrete:(a) uniaxial tension; (b) uniaxial compression.
The yield function F which consider the yield strength of tension and compression is [6,7]:
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
F (σ , ε pl )
1 (q 3 p (ε pl ) ˆ max (ˆ max )) c (c pl ) 0 1
73 3
\* MERGEFORMAT (3)
1 where and both are dimensionless material constants, p σ ：I is the effective hydrostatic pressure, σ is the 3
3 S : S is the Mises equivalent effective stress, 2 is the deviatoric part of the effective stress tensor σ , ˆ max is the algebraically maximum
effective stress tensor, ε pl is the equivalent plastic strain tensor, q where S pI σ
eigenvalue of σ ,the function ( ε pl ) is given as (ε pl )
c (cpl ) (1 ) (1 ) , where t and c are the effective t (tpl )
tensile and compressive cohesion stresses, respectively, the coefficient can be determined from the initial c0 equibiaxial and uniaxial compressive yield stress, b0 and c0 ,as b 0 . 2 b 0 c 0 The relationship between the plastic strain rate ε pl and the plastic potential function G (σ ) is: ε pl
G (σ ) σ
\* MERGEFORMAT (4)
where, 0 is a plastic multiplier. For the non-associated plastic flow rule, the plastic potential function is the Drucker-Prager hyperbolic function: 2 G(σ ) （e t0 tan） q 2 ptan
\* MERGEFORMAT (5)
where is the dilation angle, t0 is the uniaxial tensile stress at failure, e is eccentricity parameter, the flow potential tends to a straight line as the eccentricity tends to zero. 3. Damage factor determination method When the ABAQUS plastic damage constitutive model is used for analysis, the compression, tensile stress and the corresponding non-elastic strain and damage factor of the material are required. In this paper, the damage factors are calculated based on the Sidiroff’s energy equivalent principle [8]. The Sidiroff's energy equivalent principle assumes that the stress acting on the damaged material and the nondamaged material produces the same energy in the form of the same, the elastic residual energy of the non-damaged material and the equivalent elastic energy of the damaged material are: W0 e 2 （ / 2 E0）
\* MERGEFORMAT (6)
Wd e 2 （ / 2 E0） 2 （ / 2 Ed）
\* MERGEFORMAT (7)
. 1 d From formula (7),the equivalent elastic modulus is derived as Ed E0 (1 d ) 2 and the uniaxial constitutive relation and damage factor of the damaged material can be obtained: where, the effective stress is
E0 (1 d ) 2
\* MERGEFORMAT (8)
Wei Demin al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/etProcedia Engineering 00 (2017) 000–000
74 4
d 1
\* MERGEFORMAT (9)
E0
4. Verification of damage factors Elastic modulus is the main indicator of material deformation performance, determined by the relevant test [9].When the concrete material is subjected to uniaxial compression and the stress is greater than 0.3~0.5 times its compressive strength, the microfracture develops into the unstable stage, and the material exhibits obvious nonlinearity. The material is in the linear elastic phase under the action of uniaxial tension less than 0.6~0.8 times its ultimate tensile strength. Therefore, the values of the modulus of elasticity under uniaxial compression take the secant modulus at half of the working stress, and the values of the uniaxial tensile modulus of elasticity generally take the secant modulus at half of the work stress, both are equal as E0 [9]. The uniaxial tension constitutive relationship given in the norm [10] is simplified as: E x / f t y 0 t 1.7 x / [ t ( x 1) x]
( x 1)
\* MERGEFORMAT (10)
( x 1)
where y / f t , x / t , f t is the representative value of the uniaxial tensile strength of the concrete, and its value can be taken according to the actual structural analysis, respectively,the characteristic value f tk , the design value f t or the mean value f tm , t is the corresponding peak tensile strain corresponding to f t , t is the stress-strain curve descending segment parameters. The ascending and descending segments of the constitutive relation of concrete under uniaxial compression given in the norm [10] are simplified as: E0 c x / f c ( 0.5 f c ) y 2 3 a x (3 - 2 a ) x ( a 2) x (0.5 f c f c )
\* MERGEFORMAT (11)
y x / [ d ( x 1) 2 x]
\* MERGEFORMAT (12)
( x 1)
where y / f c , x / c , f c is the representative value of the uniaxial compressive strength of the concrete, and its value can be taken according to the actual structural analysis, taking the characteristic value f ck , design value f c or mean value f cm ,respectively, c is the peak pressure strain corresponding to f c , a and d are the parameters of the stress-strain curve rising and falling segments, respectively. Taking C40 grade concrete as an example to calculate the damage factor. The material properties are: uniaxial compressive strength characteristic value f ck =26.8N/mm 2 , uniaxial tensile strength characteristic value f tk =26.8N/mm 2 , elastic modulus E0 =2.7393 10 4 N/mm 2 , Poisson's ratio =0.2 , dilation angle =30 0 , eccentricity e 0.1 , the ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress is f b 0 / f c 0 1.16 , the ratio of the second stress invariant on the tensile meridian to that on the compressive
meridian K c 0.6667 , viscosity parameter 0.0001 . Table 1 is the concrete damage factors calculated from Eq. (11), (13) to (15). Table 1. C40 grade concrete damage factors. Yield stress in compression ( MPa )
Inelastic Strain (10 3 )
Compressive damage variable
Stress after cracking ( MPa )
Cracking Strain (10 3 )
Tensile damage variable
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
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13.941
0
0.000
2.390
0
0.000
24.505
0.218
0.103
2.180
0.025
0.128
26.800
0.611
0.215
1.885
0.053
0.250
24.792
1.161
0.338
1.750
0.067
0.301
21.214
1.769
0.448
1.425
0.105
0.425
18.926
2.17
0.509
1.196
0.14
0.512
15.176
2.943
0.602
0.947
0.192
0.610
10.909
4.211
0.706
0.719
0.27
0.702
6.956
6.421
0.805
0.485
0.436
0.803
5.159
8.394
0.852
0.376
0.597
0.850
3.377
12.274
0.900
0.262
0.95
0.900
3.025
13.558
0.910
0.239
1.073
0.910
2.645
15.32
0.921
0.216
1.231
0.920
2.303
17.399
0.931
0.197
1.389
0.928
1.969
20.113
0.940
1.633
23.94
0.950
1.535
25.374
0.953
Taking the standard specimen of concrete prism as an example, the element type is three-dimensional solid element C3D8R, the specimen which is divided into 54 units and the unit size is 150mm 150mm 300mm . Through the uniaxial displacement loading, the reaction force and displacement are obtained, and the relationship between the uniaxial stress and strain is calculated. The comparison between the calculated results and the norm is given in Fig.2:
(a)
(b)
Fig. 2. Uniaxial stress-strain curve: (a) Uniaxial compression; (b) Uniaxial tension.
It can be seen from Fig. 2 that the peak stress and peak strain of the concrete uniaxial constitutive curves obtained by finite element simulation are in good agreement with the code results, the peak stress and peak strain of uniaxial compression were 27.33MPa and 1.72E-3,which were 1.98% and 8.17% higher than the code value respectively. The uniaxial tensile stress and peak strain are 2.32MPa and 9.42E-5, respectively, which are 3.03% smaller than the Code value and 8.03% higher than the Code value. The calculated results show that the deviation of the stress results is smaller, and the calculated peak strain is larger than the code value because the taking value of the elastic modulus in the paper is smaller than the Code. And the other grades of concrete can be verified in the same way.
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
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5. Test simulation Petersson [11] studied the plain concrete beam shown in Fig. 3, the test results show that the nonlinear behavior of the simply supported beam caused by type I tensile cracks. The geometric dimensions are: l 2.0m , b 0.05m , h 0.2m , d 0.04m , a 0.1m .
(a)
(b)
Fig. 3. Geometric dimensions and finite element model of notch beam: (a) Geometric dimensions; (b) Finite element meshing
The simulation was carried out using the plastic damage model described above. Taking the semi-structure for analysis according to the symmetry.Using the four-node plane stress reduction integral unit CPS4R,in order to study the influence of mesh size on the analysis results, the coarse, medium and fine meshing units are used, and the total number of semi-structural elements is 352,1120,4480,respectively. The beam is loaded by prescribing the vertical displacement at the center of the beam until it reaches a value of 0.0015m and the Riks method is used. The material properties of concrete are: elastic modulus E0 30 Gpa , Poisson's ratio 0.2 , density 0 2400 kg/m3 , uniaxial tensile strength f t 3.33 MPa , fracture energy G If 124 N/m . In this paper, the tensile stiffening parameters [12] are using the stress-cracking displacement relationship based on the fracture energy cracking criterion. The linear tenion[13]softening curve, the European Model Code [14], Petersson [11] proposed bilinear tension softening curve are using for analysis, respectively, as shown in Fig. 4. The tensile damage of concrete is considered and assuming that the tensile damage factor increases linearly with the crack displacement, it reaches the maximum value until the ultimate crack displacement,is shown in Fig. 5.
Fig. 4. Softening constitutive relationship.
Fig. 5. Tensile damage factor.
Fig. 6 and Fig. 7 show the comparison of the results of finite element model analysis in different meshes. It can be seen from the figure that the results of coarse, medium and fine mesh are consistent, so the stress-cracking displacement based on fracture energy cracking criterion there is no obvious size sensitivity in response.
Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
Fig. 6. Result of linear softening constitutive model.
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Fig. 7. Result of MC bilinear softening constitutive model.
Fig. 8 shows the comparison of the results of the different tension softening constitutive models under mediummeshing and the results of the test.
Fig. 8. Comparison of simulated results of Petersson notch beam.
In the test [11], the fracture energy parameters of concrete are 115 ~ 137 N / m, the average value is 124 N / m, and the corresponding test peak load are 727.80N and 800N respectively, and the average value is 763.9N, the peak displacements are 0.334 mm and 0.396 mm, respectively, and the average value is 0.365 mm. The peak load calculated by the linear softening constitutive model, the European norm and the Petersson proposed bilinear softening constitutive model are 894.78N, 805.06N and 752.00N, respectively,the corresponding calculated displacements are 0.565mm, 0.472mm, 0.410mm. The results of single linear softening constitutive model,is too stiff compared with the experimental observations of Petersson . The calculated peak load is 17.1% higher than that of the test, and the peak displacement is 54.8% larger than that of the test,the analysis results are less accurate .The peak load calculated by the bilinear softening constitutive model proposed by the European and Petersson is 5.4% larger than the test and 1.6% smaller than the test respectively,the deviation of the load is smaller and the deviation of the displacement is larger.The results show that the bilinear softening constitutive model has better accuracy than the single linear softening constitutive model. The difference of the bilinear softening form also affects the analysis results. Fig. 9 is the contour of PEEQT, PE Max Principal, S Max Principal ,DAMAGET of the FEM ananlysis of Petersson proposed bilinear softening constitutive model in medium mesh, which are referred to as the tensile equivalent plastic strains, the maximum principal plastic strain, tensile damage ,maximum principal stress. We can assume that cracking initiates at points where the tensile equivalent plastic strain is greater than zero and the maximum principal plastic strain is positive.
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Wei Demin et al. / Procedia Engineering 210 (2017) 71–78 We Demin,He Fukang/ Procedia Engineering 00 (2017) 000–000
(a)
(b)
(c)
(d)
Fig. 9. Contour of FEM analysis: (a)PEEQT; (b) PE Max Principal; (c) DAMAGET ; (d) S Max Principal.
6. Conclusion In this paper, two kinds of tensile stiffening methods of ABAQUS plastic damage model are discussed by finite element analysis of uniaxial tension and compression of concrete and the experiment of plain concrete notch beam. Describing the strain softening characteristic of the cracking concrete based on the relationship between the subsequent failure stress and the cracking strain through the normative uniaxial constitutive relation and the damage factor determination method in this paper is available. The relevant factors of the tensile stiffening parameters and how to select the reasonable tensile stiffening parameters need further study. In terms of using the stress-cracking displacement relation based on the fracture energy cracking criterion to define the strain softening characteristics of the cracking concrete, the analysis results show that there is no obvious element size effect. The results of the bilinear softening constitutive model have better accuracy, also the influence of different softening constitutive curves on the analysis results needs further study. References [1] Yao G H, Huang Y J, Song B D, et al. Analysis for Static Behaviour of Steel and Concrete Composite Members with Plastic-Damage Model[J]. Progress in Steel Building Structures. 2009. [2] Lei T. Application of Damaged Plasticity Model for Concrete[J]. Structural Engineers. 2008. [3] G.R. Mettam, L.B. Adams, How to prepare an electronic version of your article, in: B.S. Jones, R.Z. Smith (Eds.), Introduction to the Electronic Age, E-Publishing Inc., New York, 1999, pp. 281–304. [3] Cao M. Research on Damage Plastic Calculation Method of ABAQUS Concrete Damaged Plasticity Model[J]. Transportation Standardization. 2012. [4] Zhang J, Wang Q, Hu S, et al. Parameters Verification of Concrete Damaged Plastic Model of ABAQUS[J]. Building Structure. 2008, 38(8): 127-130. [5] Qin H, Zhao X. Study on the ABAQUS Damage Parameter in the Concrete Damage Plasticity Model[J]. Structural Engineers. 2013. [6] Lubliner J, Oliver J, Oller S, et al. A plastic-damage model for concrete[J]. International Journal of Solids & Structures. 1989, 25(3): 299-326. [7] Lee J H, Fenves G L. Plastic-damage model for cyclic loading of concrete structures[J]. JOURNAL OF ENGINEERING MECHANICSASCE. 1998, 124(8): 892-900. [8] LI Zhaoxia. Damage mechanics and its applications[M]. Beijing: Science Press,2002:7-30. [9] Guo Z H.Theory of reinforced concrete[M]. Beijing:Tsinghua University Press.2013. [10] Ministry of Construction of the People’s Republic of China.GB50010-2002 Code for design of concrete structures[M]. Beijing:China Architecture & Building Press.2002. [11] Petersson P E. Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials[J]. Report No.TVBM1006,Division of Building Materials,University of Lund,Sweden. [12] Wang J C,Chen Y K. Application of ABAQUS in Civil Engineering[M].Hangzhou: Zhejiang University Press.2006. [13] Wang Q Y. Research on concrete softening relationship from notched three-point bending beam based on evolutionary algorithms[D]. Hangzhou: Zhejiang University of Technology.2012. [14] Betonbau. fib Model Code for Concrete Structures 2010[J]. Ernst & Sohn. 2013.