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Three dimensional combined fracture–plastic material model for concrete ˇ ervenka a, Vassilis K. Papanikolaou b,* Jan C Cˇervenka Consulting, Predvoje 22, 16200 Prague 6, Czech Republic Laboratory of Reinforced Concrete and Masonry Structures, Civil Engineering Department, Aristotle University of Thessaloniki, P.O. Box 482, Thessaloniki, 54124, Greece a
b
Received 30 July 2007; received in final revised form 23 January 2008
Abstract This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Mene´trey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: A. Fracture; B. Concrete; B. Constitutive behavior; B. Elastic–plastic material; C. Finite elements
*
Corresponding author. Tel.: +30 2310995662; fax: +30 2310995614. E-mail address:
[email protected] (V.K. Papanikolaou).
0749-6419/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2008.01.004
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Nomenclature a A B C c(j) dk D, Dijkl Ds 0 D cr e ep Ec ff fp fc ft g Gf H k(j) ko lij Lt m n nk n1 n2 r(h, e) rg t T w wo af ap b e, eij ee, eeij ep, epij ef, efij ^efk epv
attraction parameter of the plastic potential function first coefficient of the plastic potential function second coefficient of the plastic potential function third coefficient of the plastic potential function softening function plastic and fracture multiplier elasticity matrix secant constitutive matrix cracked stiffness matrix eccentricity parameter of the Mene´trey–Willam failure surface plastic deviatoric strain concrete elastic modulus rankine failure surface (fracture) Mene´trey–Willam failure surface (plasticity) uniaxial compressive concrete strength uniaxial tensile concrete strength plastic potential function fracture energy hardening modulus hardening function hardening parameter defining the onset of plastic flow stress return direction characteristic length friction parameter of the Mene´trey–Willam failure surface order of the plastic potential function eigenvector defining direction k first parameter of softening function second parameter of softening function elliptic function of the Mene´trey–Willam failure surface crack shear stiffness coefficient slope parameter of the softening function transformation matrix crack opening crack opening corresponding to zero tensile stress fracture contribution in the combination algorithm plasticity contribution in the combination algorithm relaxation factor in the combination algorithm total strain vector and tensor elastic strain vector and tensor plastic strain vector and tensor fracture strain vector and tensor maximum fracture strain in direction k plastic volumetric strain
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epv;t h j kt m n n0 q q0 r, rij rco v w
3
plastic volumetric strain at uniaxial concrete strength (threshold value) lode angle of the stress vector in Haigh–Westergaard stress space hardening/softening parameter scaling factor for the tensile strength ft of the plasticity model concrete Poisson’s ratio hydrostatic length of the stress vector in Haigh–Westergaard stress space hydrostatic length of the plastic strain vector deviatoric length of the stress vector in Haigh–Westergaard stress space deviatoric length of the plastic strain vector stress vector and tensor concrete stress defining the onset of plastic flow convergence rate of the combination algorithm inclination of the plastic strain vector
1. Introduction This paper describes a three dimensional constitutive material model for concrete, which combines plasticity with fracture. Fracture is modeled by an orthotropic smeared crack model based on the Rankine tensile criterion. A hardening/softening plasticity model based on the Mene´trey and Willam (1995) three-parameter failure surface is used to simulate concrete crushing. Although many papers have been published on plasticity models for concrete (e.g. Pramono and Willam, 1989; Etse, 1992; Feenstra, 1993; Mene´trey et al., 1997; Feenstra et al., 1998; Grassl et al., 2002) or smeared crack models ˇ ervenka and Gerstle, 1971; Bazˇant and Oh, 1983; De Borst, (e.g. Rashid, 1968; C 1986; Rots and Blaauwendraad, 1989), there are not many descriptions of their successful combination in the literature. Owen et al. (1983) presented a combination of cracking and visco-plasticity. Comprehensive treatise of the problem was provided also by De Borst (1986), and several works have been published on the combination of damage and plasticity (e.g. Simo and Ju, 1987; Meschke et al., 1988; Bielger and Mehrabadi, 1995; Lee and Fenves, 1998; or more recent works by Grassl and Jira´sek (2006), Mohamad-Hussein and Shao (2007), Contrafatto and Cuomo (2006), Jason et al. (2006), Cicekli et al. (2007) and Chiarelli et al. (2003) for rock material). Various concrete models have been proposed in the literature that are based on different approaches such as for instance the class of microplane models (Bazˇant et al., 2000) or models based on concrete micromechanics (Mattei et al., 2007), which are derived using the theory of granular materials (Christoffersen et al., 1981; Oda et al., 1982; Mehrabadi et al., 1982). The plastic-damage models are usually formulated within the concept of thermodynamics, and with the exception of the works by Meschke et al. (1988) and Cicekli et al. (2007), they usually consider an isotropic damage formulation, which neglects the anisotropic nature of cracked concrete behavior. In the presented model, the cracked concrete is modeled as an orthotropic material and it considers the problematic of physical changes, like for instance crack closure as it is the case of the model proposed by Cicekli et al. (2007). In addition, it considers the shear behavior of cracked concrete and rotated as well as fixed crack formulation. Also within the proposed approach it is possible to formulate both ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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models (i.e. plastic and fracture) entirely separately, and their combination can be provided in a separate algorithm. From a programming point of view, such an approach is well suited for object oriented programming. ˇ erThe suggested model is an improved version of a previously published model by C venka et al. (1998) that includes various enhancements in both fracturing and plasticity parts. The fracturing part of the model have been extended to address the mode II and III (i.e. shear) crack propagation. The plasticity part is extended to include the formulations suggested by Papanikolaou and Kappos (2007), which describe the increased strength and deformation capacity of concrete under multiaxial compression and can properly handle problems including confinement effects. The major difference between the present model and the one suggested by the above writers is that the former (using a special algorithm that combines plasticity with fracture) can efficiently handle all possible loading and unloading paths (tension, compression or combinations of both) and describe crack opening and closure as well. Another advantage of the present model is that it was successfully integrated in a general finite element package (ATENA, ˇ ervenka et al., 2007) and can be directly applied to the analysis of complex reinforced C concrete structures, rather than simulating only multiaxial loading of plain concrete (using a constitutive driver). The approach of strain decomposition introduced by De Borst (1986) is used to combine fracture and plasticity models together. Both models are developed within the framework of the return-mapping algorithm (Wilkins, 1964). This approach guarantees the solution for all magnitudes of strain increment. From an algorithmic point of view, the problem is then transformed into finding an optimal return point on the active failure surfaces. The combined algorithm must determine the separation of strains into plastic and fracturing components, while it must preserve stress equivalence in both models. The proposed algorithm is based on a recursive iterative scheme. It can be shown that such a recursive algorithm cannot reach convergence in certain cases such as, for instance, softening and dilating materials. For this reason the recursive algorithm is extended by a relaxation method, in order to stabilize convergence. In the first part of the paper, the constitutive equations of the fracture and plastic model are presented. This part also contains a description of the recursive algorithm for the combination of the two material models. In the subsequent section, the numerical behavior of this algorithm is demonstrated under several selected loading histories, along with comparisons with experimental results. The last section demonstrates the model performance in practical engineering applications. 2. Material model formulation The material model formulation assumes small strains and is based on the strain decomposition into elastic ðeeij Þ, plastic ðepij Þ and fracture ðefij Þ components (De Borst, 1986), which can be written in the rate form as e_ ij ¼ e_ eij þ e_ pij þ e_ fij
ð1Þ
The stress development can be then described by the following rate equations describing the progressive degradation (concrete cracking) and plastic yielding (concrete crushing): r_ ij ¼ Dijkl ð_ekl e_ pkl e_ fkl Þ
ð2Þ
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where the fracture strain rate ð_efij Þ and plastic strain rate ð_epij Þ are evaluated from the fracture and plasticity models respectively. The constitutive equations of the both models can be summarized as follows: Flow rule governs the evolution of plastic and fracturing strains: Plastic model :
e_ pij ¼ k_ p mpij ;
mpij ¼
ogp orij
ð3Þ
Fracture model :
e_ fij ¼ k_ f mfij ;
mfij ¼
ogf orij
ð4Þ
where k_ p is the plastic multiplier rate and gp is the plastic potential function. Following the unified theory of elastic degradation of Carol et al. (1994) it is possible to define analogous quantities for the fracturing model, i.e. k_ f is the inelastic fracturing multiplier, respectively, and gf is the potential defining the direction of inelastic fracturing strains in the fracturing model. The consistency conditions can be than used to evaluate the change of the plastic and fracturing multipliers. f_ p ¼ npij r_ ij þ H p k_ p ¼ 0;
npij ¼
of p orij
ð5Þ
f_ f ¼ nfij r_ ij þ H f k_ f ¼ 0;
nfij ¼
of f orij
ð6Þ
This represents and system of two equations for the two unknown multiplier rates k_ p and k_ f , and is analogous to the problem of multi-surface plasticity (Simo et al., 1988). The combination of the two models is described in detail in Section 2.3. 2.1. Fracture model for concrete cracking The Rankine criterion is used for concrete cracking. For each direction (k = 1, 2, 3), it is expressed as fkf ¼ t rij nki nkj ft 6 0
ð7Þ
or in Haigh–Westergaard coordinates (Fig. 1):
Fig. 1. Rankine failure surface represented in Haigh–Westergaard coordinates: (a) 3D stress space, (b) Rendulic plane and (c) deviatoric plane.
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ff ¼ n
pffiffiffi pffiffiffi 2 q cos h 3 ft 6 0
ð8Þ
The Rankine criterion (7) represents actually three distinct planes forming a pyramid in stress space, as depicted in Fig. 1. It is assumed that strains and stresses are converted into material directions given by the eigenvectors nk, which in the case of rotated crack model correspond to the instantaneous principal strain directions, and in the case of fixed crack model are given by the principal strain directions at the onset of cracking. Therefore, trij identifies with the trial stress and ft with the concrete tensile strength. The trial stress state is computed by the elastic predictor: t
rij ¼ n rij þ Dijkl dekl
ð9Þ
t
If the trial stress ( rij) does not satisfy the Rankine failure criterion (Eq. (7)), the increment of fracture strain in direction k can be calculated using the assumption that the final stress state must satisfy the following equation: fkf ¼ nþ1 rij nki nkj ft ¼ t rij nki nkj Dijmn defmn nki nkj ft ¼ 0
ð10Þ
This equation can be further simplified under the assumption that the increment of fracturing strain is normal to the Rankine failure surface, and that always only one direction is being checked each time. For failure surface k, the fracturing strain increment has the following form (associated form, i.e. gf = ff): k de0fk ij ¼ dk
ofkf ¼ dkk nki nkj orij
ð11Þ
After substituting Eq. (11) in Eq. (10), a formula for the increment of the fracturing multiplier is recovered: t
dkk ¼
rij nki nkj ft ðwk Þ Dijmn nki nkj nkm nkn
ð12Þ
where wk ¼ Lt ð^efk þ dkk Þ
ð13Þ
The system of Eqs. (12) and (13) must be solved iteratively since for softening materials the value of the current tensile strength ft (wk) is a function of the crack opening (wk) which is based on the following empirical formula suggested by Hordijk (1991): " 3 # r w w c w ¼ 1 þ c1 ð1 þ c31 Þ ec2 ð14Þ e 2 wo ft wo wo where: r is the tensile concrete stress normal to crack, ft is the concrete tensile strength, c1 = 3, c2 = 6.93 and wo = 5.14 Gf/ft. (Gf is the fracture energy of the material, provided as a model parameter, see Table 1.) The crack opening (w) is computed from the total accumulated value of fracturing strain ð^efk Þ in direction k, plus the current increment of fracturing strain (dk), and this sum is multiplied by the characteristic length (Lt). The characteristic length as a crack band size was introduced by Bazˇant and Oh (1983). Various methods were proposed for the crack band size calculation in the framework of the finite element method. Feenstra (1993) suggested a method based on integration point volume, which is not well suited for ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
20
30
40
50
60
70
80
90
100
110
120
Ec (MPa) m ft (MPa) kt e fco (MPa) epv;t t A B C n Gf (MN/m)
24377 0.2 1.917 1.043 0.5281 4.32 4.92 104 1.33 103 7.342177 8.032485 3.726514 3 4.87 105
27530 0.2 2.446 1.227 0.5232 9.16 6.54 104 2.00 103 5.436344 6.563421 3.25626 3 6.47 105
30011 0.2 2.906 1.376 0.5198 15.62 8.00 104 2.67 103 4.371435 5.73549 3.055953 3 7.92 105
32089 0.2 3.323 1.505 0.5172 23.63 9.35 104 3.33 103 3.971437 5.430334 2.903173 3 9.26 105
33893 0.2 3.707 1.619 0.5151 33.14 1.06 103 4.00 103 3.674375 5.202794 2.797059 3 1.05 104
35497 0.2 4.066 1.722 0.5133 44.11 1.18 103 4.67 103 3.43856 5.021407 2.719067 3 1.17 104
36948 0.2 4.405 1.816 0.5117 56.50 1.30 103 5.33 103 3.245006 4.871993 2.659098 3 1.29 104
38277 0.2 4.728 1.904 0.5104 70.30 1.41 103 6.00 103 3.082129 4.745867 2.611426 3 1.40 104
39506 0.2 5.036 1.986 0.5092 85.48 1.52 103 6.67 103 2.942391 4.637358 2.572571 3 1.50 104
40652 0.2 5.333 2.063 0.5081 102.01 1.62 103 7.33 103 2.820644 4.542587 2.540158 3 1.61 104
41727 0.2 5.618 2.136 0.5071 114.00 1.73 103 8.00 103 2.713227 4.458782 2.512681 3 1.71 104
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Table 1 Suggested parameters for fracture and plasticity models
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distorted elements. A consistent and rather complex approach was proposed by Olivier (1989). In the present model, the crack band size is calculated as a width or size of the eleˇ ervenka et al. (1995) showed that this ment projected into the direction k (Fig. 2). C approach is satisfactory for low order linear finite elements, which are used throughout this study. They also proposed a modification, which accounts for cracks that are not aligned with element edges. The crack band approach assures that the energy dissipation does not depend on the finite element size. Other methods have been proposed in the literature based on nonlocal averaging (Bazˇant and Pijaudier-Cabot, 1987) or gradient approaches (De Borst and Mu¨hlhaus, 1992), and a recent work by Hashiguchi and Tsutsumi (2007). The system of Eqs. (12) and (13) can be solved by recursive substitutions. It is possible to show by expanding ft(wk) into a Taylor series that this iteration scheme converges as long as oft ðwk Þ Dijmn nki nkj nkm nkn < ð15Þ ow Lt Eq. (15) is violated for softening materials only when snap back is observed in the stress–strain relationship, which can occur if large finite elements are used. In the standard displacement based finite element method, the strain increment is given and therefore, a snap back on the constitutive level cannot be captured. This means that the critical region, a snap back on the softening curve will be skipped in a real calculation, which physically means that the energy dissipated by the system will be overestimated. This is of course minðD Þ undesirable, and finite elements smaller than Lt < of ð0Þkkkk should be used, where t ow oft ð0Þ ow denotes the initial slope of the crack softening curve. It is important to distinguish between the total fracturing strain ð^efk Þ, which corresponds to the maximum fracturing strain at material direction k reached during the loading process, and the current fracturing strain ðefij Þ, which can be smaller due to crack closure. The local fracturing strains e0fij in the local coordinate system given by the material directions nk can be calculated using the following equation derived by Rots and Blaauwendraad (1989) in the matrix form: e0f ¼ ðD þ D0cr Þ
1
De
ð16Þ
Fig. 2. Tensile strength function with respect to crack width (Hordijk, 1991).
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where D0cr ijkl is the stiffness inside the crack zone defined as 0f r0ij ¼ D0cr ijkl ekl
ð17Þ D0cr ijkl
The fourth order crack tensor represents the cracking stiffness in the local material directions. In the current formulation, it is assumed that there is no interaction between normal and shear components. Thus, the nonzero crack tensor elements are given by the following formulas: Mode I (normal) crack stiffness: D0cr kkkk ¼
ft ðwk Þ ^efk
ðno index summationÞ
ð18Þ
Mode II and III (shear) crack stiffness: 0cr 0cr g D0cr ijij ¼ minðDiiii ; Djjjj Þ r
ðno index summation and i 6¼ jÞ
ð19Þ
g
where r is the crack shear stiffness coefficient, which is an input parameter. It represents the ratio of the shear crack stiffness to the normal crack stiffness. The recommended value is within the range of 1–10. In Eqs. (18) and (19), it is necessary to handle the special cases before the onset of cracking, when the expressions approach infinity. Large penalty numft ð0Þ bers are used for normal crack stiffness in these cases: D0cr kkkk ¼ e , where e is a small number (also ^efk is initialized to e at the beginning). The secant constitutive matrix in the material directions can be derived from Eq. (2) using Eq. (16): D0s ¼ D D ðD0cr þ DÞ
1
D
ð20Þ
Strain vector transformation matrix T (i.e. global to local strain transformation matrix) can be used to transform the local secant stiffness matrix to the global coordinate system. A detailed flow chart of the fracture model is shown in Fig. 3. Ds ¼ TT D0s T
ð21Þ
2.2. Plasticity model for concrete crushing The new stress state in the plastic model is computed using the predictor-corrector formula: nþ1
rij ¼ n rij þ Dijkl ðdekl depkl Þ ¼ t rij Dijkl depkl ¼ t rij rpij
ð22Þ
The plastic corrector ðrpij Þ is computed directly from the yield function by the return-mapping algorithm: f p ðt rij rpij Þ ¼ f p ðt rij dk lij Þ ¼ 0
ð23Þ
The crucial aspect is the definition of the return direction (lij) which can be defined as lij ¼ Dijkl
ogðt rkl Þ orkl
ð24Þ
where g(trij) is the plastic potential function, which derivative is evaluated at the predictor stress state (trij) to determine the return direction. ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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Fig. 3. Flow chart of the fracture model for concrete cracking.
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Fig. 4. Mene´trey–Willam failure criterion represented in Haigh–Westergaard coordinates: (a) 3D stress space, (b) Rendulic plane and (c) deviatoric plane.
In the present plasticity model, the Mene´trey and Willam (1995) three parameter failure surface is used (Fig. 4): pffiffiffiffiffiffiffi f ðn;q;hÞ ¼ 1:5 p
q kðjÞ fc
2
q
n
!
þ m pffiffiffi rðh;eÞ þ pffiffiffi cðjÞ ¼ 0 6 kðjÞ fc 3 kðjÞ fc ð25Þ
where:
m¼3
ðkðjÞ fc Þ2 ðkt ft Þ2 e is the cohesion parameter of the material and kðjÞ fc kt ft eþ1
rðh;eÞ ¼
4ð1 e2 Þ cos2 h þ ð2e 1Þ2 2ð1 e2 Þcos h þ ð2e 1Þ½4ð1 e2 Þ cos2 h þ 5e2 4e1=2
ð26Þ
is an elliptic function ð27Þ
In the above equations, (n, q, h) are the Haigh–Westergaard coordinates and fc and ft are the compressive strength and tensile strength respectively. Parameter kt > 1 is a scaling value for the tensile concrete strength in order to provide intersection between the Rankine (fracture) and the Mene´trey–Willam (plasticity) failure surfaces during the combination procedure (Fig. 5). Parameter e 2 (0.5, 1.0) defines the roundness of the Mene´trey–Willam failure surface, with a recommended value e = 0.52 (Mene´trey and Willam, 1995) leading to equibiaxial concrete strength equal to fbc = 1.14 fc, very similar to the experimental work by Kupfer et al. (1969). The position of the Mene´trey–Willam failure surface is not fixed but it can expand and move along the hydrostatic axis (simulating the hardening and softening stages), based on the value of the hardening/softening parameter (j). In the current model, this parameter identifies with the volumetric plastic strain (Grassl et al., 2002): dj ¼ depv ¼ dep1 þ dep2 þ dep3
ð28Þ
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Fig. 5. (a) Intersection of Rankine and Mene´trey–Willam failure surfaces for kt = 2 and (b) no intersection for kt = 1.
The instantaneous shape and location of the loading surface during hardening is defined by a hardening function (k), which depends on the hardening/softening parameter (j). This function is directly incorporated in the Mene´trey–Willam failure surface Eqs. (25) and (26), operating as a scaling factor on the compressive concrete strength (fc). It has the ˇ ervenka et al., 1998): following elliptic form (C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 ev;t epv ð29Þ kðjÞ ¼ kðepv Þ ¼ k o þ ð1 k o Þ 1 epv;t where epv;t is the plastic volumetric strain at uniaxial concrete strength (onset of softening) and ko is the value that defines the initial yield surface that bounds the initial elastic regime (onset of plasticity). At the end of the hardening process, the hardening function retains a constant value of unity and the material enters the softening regime, which is controlled by the softening function (c). This function simulates the material decohesion by shifting the ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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loading surface along the negative hydrostatic axis. It is assumed that it follows the softening function originally proposed by Van Gysel and Taerwe (1996) for uniaxial compression: 0 12 B cðjÞ ¼ cðepv Þ ¼ @
1þ
1
n1 1 n2 1
C 2 A
ð30Þ
where: n1 ¼ epv =epv;t n2 ¼
ðepv;t
þ
ð31Þ tÞ=epv;t
ð32Þ
Parameter t in Eq. (32) controls the slope of the softening function and the outmost square is necessary due to the quadratic nature of the loading surface. The softening function value starts from unity and complete material decohesion is attained at c = 0. The evolution of both hardening and softening functions with respect to the hardening/softening parameter is schematically shown in Fig. 6. The plasticity model incorporates a non-associated flow rule using a polynomial plastic potential function (g), with Lode angle (h) dependency and adjustable order (n): n q 1 q n pffiffiffi g ¼A þ C þ ðB CÞð1 cos 3hÞ pffiffiffi þ pffiffiffi a 2 k c fc k c fc k c fc ð33Þ Parameters A, B and C define the shape of the plastic potential function in stress space and their calibration is based on the assumption that the inclination (w) of the incremental plastic strain vector identifies with the inclination of the total plastic strain vector at three distinct stress states, namely the uniaxial, equibiaxial and triaxial compressive concrete k(κ ) / c(κ )
k
c
1.0
k 0.8
0.6
c
0.4
0.2
0.0
ko
p
ε v,t
κ = εpv
Fig. 6. Evolution of hardening (k) and softening (c) functions with respect to the plastic volumetric strain.
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Fig. 7. Direction (w) of the incremental (a) and total (b) plastic strain vectors.
strength (Fig. 7). The presented model adopts a regular flow rule in a sense that the plastic flow direction depends solely on the location on the yield surface, and is not affected by the loading direction (Nicot and Darve, 2007). A detailed calibration scheme for the plasticity model parameters, based on an extensive experimental database can be found in Papanikolaou and Kappos (2007) and suggested values (including the fracture model parameters) for various uniaxial compressive concrete strengths (fc) are shown in Table 1. For the integration of the constitutive equations, an implicit backward-Euler returnˇ ermapping algorithm was applied, incorporating a regula-falsi/secant iterative scheme (C venka et al., 1998). In order to maintain consistency with constitutive equations, the above scheme iteratively updates the following variables: (1) the plastic multiplier (dk) which yields both the stress corrector ðdkDijkl mkkl Þ and the additional plastic strains ðþdkmkij Þ, (2) the return direction ðmkij Þ of the stress vector towards the Mene´trey–Willam failure surface and (3) the hardening/softening functions (k(j), c(j)), which control the position and the size of the Mene´trey–Willam failure surface (f) and the plastic potential surface (g) in the stress space. The input data (step n) are the current stress tensor (nrij), the plastic strain tensor ðn epij Þ, describing the deformation history (load path dependency) of the material, and the trial total strain increment (deij). The output data (step n + 1) are the updated nonlinear stress tensor (n + 1rij) and the updated plastic strain tensor ðnþ1 epij Þ. The suggested algorithm is numerically stable, independent of the load step size and does not require the differentiation of the Mene´trey–Willam failure surface. A detailed flow chart of the procedure is shown in Fig. 8 and a schematic depiction of the stress return process is shown in Fig. 9. 2.3. Combination of plasticity and fracture models The objective is to combine the above models into a single model so that the plasticity model is used for concrete crushing and the fracture model for cracking. This problem can be generally stated as a simultaneous solution of the two following inequalities: ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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Fig. 8. Flow chart of the backward-Euler return-mapping algorithm.
Fig. 9. Schematic depiction of the return process.
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f p ðn rij þ Dijkl ðdekl defkl depkl ÞÞ 6 0 ! solve for depkl f n
f ð rij þ Dijkl ðdekl
depkl
defkl ÞÞ
6 0 ! solve for
ð34Þ
defkl
ð35Þ
Each of the above expressions depends on the output from the other, and therefore the following iterative scheme is applied: fði1Þ
Step 1 : f p ðn rij þ Dijkl ðde kl dekl
corði1Þ
þ b dekl
pðiÞ
fðiÞ
pðiÞ
pðiÞ
dekl ÞÞ 6 0 ! solve for dekl ð36Þ fðiÞ
Step 2 : f f ðn rij þ Dijkl ðdekl dekl dekl ÞÞ 6 0 ! solve for dekl Step 3 :
corðiÞ deij
¼
fðiÞ deij
fði1Þ deij
ð37Þ ð38Þ
Iterative correction of the strain norm between two subsequent iterations is expressed as corðiÞ
kdeij
corði1Þ
k ¼ ð1 bÞ af ap kdeij
k
ð39Þ
where: fðiÞ
af ¼
pðiÞ kdeij pðiÞ
ap ¼
fði1Þ
kdeij deij
k
ð40Þ
pði1Þ deij k pði1Þ
kdeij deij
k
ð41Þ
corði1Þ k kdeij
and b is a relaxation factor, which is introduced in order to guarantee convergence. Its calculation is based on the runtime values of af and ap, so that the convergence of the iterative scheme can be assured. The parameters af and ap characterize the mapping properties of each model (i.e. plastic and fracture). It is possible to consider each model as an operator, which maps strain increment on the input into a fracture or plastic strain increment on the output. The product of the two mappings must be contractive in order to guarantee convergence. The necessary condition for convergence is ð1 bÞ af ap < 1
ð42Þ f
p
It can be shown that the values of a and a are directly proportional to the softening rate in each model. Since the softening model remains usually constant for a material model and a finite element, their values do not change significantly between iterations. It is possible to select the scalar b, such that Eq. (42) is always satisfied at the end of each iteration, based on the current values of af and ap. There are three possible scenarios, which must be handled, for the appropriate calculation of b: ð1Þ
jaf ap j 6 v
ð43Þ
where v is related to the requested convergence rate. For a linear rate it can be set to v = 1/ 2. In this case the convergence is satisfactory and b is set to zero. v < jaf ap j < 1
ð44Þ jaf ap j then the convergence would be slow. In this case b should be calculated as b ¼ 1 v , in order to increase the convergence rate. ð2Þ
1 6 jaf ap j
ð45Þ v then the algorithm is diverging. In this case b should be calculated as b ¼ 1 f p to ja a j stabilize the iterations. ð3Þ
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This approach guarantees convergence as long as the parameters af and ap do not change drastically between iterations which is satisfied for smooth and correctly formulated finite element models. The rate of convergence depends on material brittleness, dilation rate and finite element size. It is advantageous to further stabilize the algorithm by smoothing the relaxation factor (b) during the iterative process: b¼
bi þ bi1 2
ð46Þ
where the superscripts i and i 1 denote values from two subsequent iterations. This will eliminate problems due to possible oscillation of the relaxation factor. An important condition for the convergence of the above algorithm is that the failure surfaces of the two models (Rankine and Mene´trey–Willam) are intersecting each other in all possible positions during hardening and softening, which is provided by parameter kt (Fig. 5). Additional constraints are used in the iterative algorithm; if the stress state at the end of the first step violates the Rankine criterion, the order of the first two steps of the algorithm (36 and 37) is reversed. Also in reality concrete crushing in one direction has an effect on the cracking in other directions. It is assumed that after the plasticity yield criterion is violated, the tensile strength in all material directions is gradually reduced to zero. The proposed algorithm for the combination of plastic and fracture models is schematically shown in Fig. 10. When both surfaces are activated, the behavior is quite similar to the multi-surface plasticity (Simo et al., 1988). However, contrary to the multi-surface plasticity algorithm, the proposed method is more general in the sense that it covers all loading regimes including physical changes such as crack closure. Currently, it is developed only for two interacting models, and its extension to multiple models is not straightforward.
Fig. 10. Schematic description of the iterative process shown in two dimensions for clarity.
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2.4. Model verification A constitutive driver was developed for testing various stress–strain loading histories in order to investigate the behavior of the proposed model. In this section, the constitutive driver is used to verify the numerical behavior of the proposed algorithm for the combination of the fracture and plastic models. Three distinct loading scenarios were considered: (1) Uniaxial compression beyond the maximum compressive concrete strength (crushing), then unloading into tension and then reloading again into compression. In Fig. 11 it is shown that during unloading, a crack opens in the loading direction 3 just after the attainment of concrete tensile strength (which has been reduced due to earlier crushing) and eventually closes when the loading is reversed. The material response thereafter follows the same uniaxial compression load path, just before unloading took place. (2) Crack closure due to crushing in another material direction (Fig. 12). In this example a crack is first introduced in direction 1. After the crack has fully opened and no more stresses can be transferred across the crack, displacements are constrained in direction 1 and compressive strains are applied in direction 3. After crushing starts in direction 3, the plastic strains in direction 1 will ultimately close the crack. After the crack in direction 1 closes, a biaxial compression state is recovered. At this point, stress r3 should further increase until it reaches a (second) softening branch, which corresponds to the appropriate biaxial stress state. (3) In the third example, loading is applied in all three material directions (Fig. 13). Equivalent magnitudes of tensile strain increments are first applied in directions 1 and 2, introducing two cracks respectively. Displacements are then constrained in directions 1 and 2 and compressive strains are applied in direction 3. This means that all three surfaces and both models are active. After crushing in direction 3, both cracks will close and a triaxial compression state will be recovered. fc= 30 MPa
-35
ft= 2.45 MPa
fc
3
-30 -25 -20 -15 -10 reduced tensile strength due to crushing
-5 crack opens 0
ft
3
5 0
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
Fig. 11. Example of concrete uniaxial compression-tension-compression load path.
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fc= 30 MPa
-35
ft= 2.45 MPa
biaxial stress state
fc
3
19
-30 crack closes
-25 -20 -15 -10
First load stage : direction 1 (x) Second load stage : direction 3 (z)
-5 crack opens 0 5
ε1
ft 0.002
3
0
-0.002
-0.004
-0.006
-0.008
Fig. 12. Example of concrete biaxial stress state, introduced by crack closure due to concrete crushing.
fc = 30 MPa ft = 2.45 MPa
-90
trixial stress state
3
-80 -70
First load stage : directions 1 and 2 (x and y) -60
Second load stage : direction 3 (z)
-50 -40
fc
-30 both cracks close -20 -10 both cracks open 0 10 0.003
1
=
ft
2 0.002
0.001
0.000
3 -0.001
-0.002
-0.003
-0.004
-0.005
-0.006
Fig. 13. Example of concrete triaxial stress state, introduced by crack closure due to concrete crushing.
The successful treatment of the above complex loading scenarios by the combination algorithm validates its numerical stability in the context of the present material model. In order to assess the accuracy of its performance, the model was further evaluated against several experimental results from the literature on uniaxial, biaxial and triaxial tests (Figs. 14–20). The correlation is reasonable, which renders the model applicable to large scale finite element analysis of plain and reinforced concrete structures. ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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20
-120
3 (MPa)
fc = 105.7 MPa
-100
93.9
-80
65.0 -60
50.3 -40
31.7 21.7
-20 3
0 0.000
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
Fig. 14. Comparison between numerical and experimental results on uniaxial compression (Dahl, 1992).
1 (MPa)
3.2 (
2
= 0.55 3
2.8
3
ft = 2.6 MPa
1)
(
2
= 0.55
(
1)
2, 3
(
2
= 0.55
2
= 0)
2
2=
(
1)
1
1, 2
1
1)
2.4 (
2=
(
1)
2
= 0)
2.0 1.6 : tensile : tensile 3 =0 1 2
1.2 0.8 2
= 0 : uniaxial extension 1 : equibiaxial extension
2=
0.4
(+)
(-) 0.0 -0.00005
0.00000
0.00005
0.00010
Fig. 15. Comparison between numerical and experimental results on uniaxial and biaxial tension (Kupfer et al., 1969).
3. Applications in finite element analysis The suggested model was integrated in the general finite element package ATENA ˇ ervenka et al., 2007) and was applied on three dimensional analysis of reinforced (C ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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3 (MPa)
21
fc = 32.0 MPa
-45 3
2
1
-40 ( -35
1
2
= 0.52
2, 3
(
2=
(
2
1)
1)
-30 1, 2
-25 3
-20 1
=0
= 0)
: compressive 3 : compressive
2
-15 -10 2
= 0 : uniaxial compression 1 : equibiaxial compression
2=
-5
(+)
(-)
0 0.005
0.003
0.001
-0.001
-0.003
-0.005
Fig. 16. Comparison between numerical and experimental results on biaxial tension-compression (Kupfer et al., 1969).
3 (MPa)
-35 1, 2
fc = 32.0 MPa (
3
: tensile =0 3 : compressive 1
1
= 0)
2
-30
-25 3
-20 ( 1/
1
3
= -0.1)
2
-15 1
2
3
( 1/
3
= -0.2)
-10
-5
(+) 0 0.0015
(-) 0.0010
0.0005
0.0000
-0.0005
-0.0010
-0.0015
-0.0020
-0.0025
Fig. 17. Comparison between numerical and experimental results on biaxial compression (Kupfer et al., 1969).
concrete structures. In the first example, the well known Leonhardt’s shear beam (Leonhardt and Walther, 1962) was modeled using three dimensional solid elements for concrete and embedded linear truss elements for steel. This structural element was selected because - according to experimental evidence - it exhibits both tensile and compressive stress states ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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fc = 28.6 MPa
3 (MPa)
-120
1=
2
= 21
Pa
14.7
Pa
-100
-80
8.4
-60
4.2
-40
2.1 1.05
-20
1= 0 0.04
Pa
Pa Pa
2 0.03
Pa
3 0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
Fig. 18. Comparison between numerical and experimental results on triaxial compression of normal strength concrete (Imran, 1994).
3 (MPa)
fc = 60.2 MPa
-200 1 = 2 = 29.3
Pa
23.3
Pa
20.3
Pa
14.3
Pa
11.3
Pa
8.3
Pa
-160
-120
-80
5.3 2.3
-40
0
Pa
Pa
3
Pa
0 0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
Fig. 19. Comparison between numerical and experimental results on triaxial compression of high strength concrete (Xie et al., 1995).
together with shear failure, which is expected to activate both parts (fracture and plasticity) of the present combined constitutive model. Fig. 21 shows the model geometry, material properties and solution parameters. Due to symmetry, half length of the beam was modeled using proper boundary conditions on the symmetry plane. Fig. 22 shows a comparison between numerical and experimental results (applied vertical force versus vertical deflection at midspan), the crack development (starting from small diffused flexural cracks ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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near the midspan which concludes to a large diagonal crack between loading and support points) and the embedded reinforcement axial stresses. It is observed that correlation is very reasonable concerning both the load–displacement response and the failure mode (shear failure). The second example exhibits the confinement sensitivity of the present model under compressive loading. A confined square reinforced concrete column, tested experimen-
fc = 60.6 MPa
3 (MPa)
-140
1=
2
= 12
Pa
-120
8 MPa
-100
-80
4
Pa
-60
0 MPa -40
-20
1= 0 0.015
2
3 0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
Fig. 20. Comparison between numerical and experimental results on triaxial compression of high strength concrete (Candappa et al., 2001).
Concrete strength : 28.5 MPa Number of brick elements : 5416 Number of truss elements : 2 Prescribed displacements with Newton-Raphson solution
Model geometry (half length due to symmetry)
Finite Element Mesh
Fig. 21. Three dimensional finite element model of the shear beam (Leonhardt and Walther, 1962).
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Failure at 0.0033 m (d)
80
Applied force (kN) 70
(c) 60 50
(b) 40
(a)
30
Analysis
(a) Max principal strains and cracks @ 0.0005 m
Experiment
20 10
Vertical deflection @ midspan (m) 0 0
0.001
0.002
0.003
Applied force - vertical deflection at midspan
0.004
(b) Max principal strains and cracks @ 0.001 m
Experimental shear failure mode
(c) Max principal strains and cracks @ 0.002 m
Embedded reinforcement axial stresses @ 0.0033 m
(d) Max principal strains and cracks @ 0.0033 m
Fig. 22. Nonlinear response and crack patterns of the shear beam.
tally by Sheikh and Uzumeri (1980) is modeled using solid elements for concrete and embedded truss elements for longitudinal and transverse reinforcement (Papanikolaou and Kappos, 2005). Due to double symmetry, one quarter of the column was modeled, the appropriate boundary conditions were applied on the symmetry planes and concentric compressive displacements were prescribed. Fig. 23 shows the concrete finite element mesh, the embedded reinforcement bars and the model loading and boundary conditions. Fig. 24 shows a comparison between numerical and experimental results. Reasonable correlation is observed concerning both the maximum strength (2.4% difference) and the post-peak response. Moreover, the experimentally observed cracking of the concrete cover was successfully captured by the analysis. It has to be noted here that it is necessary that the finite element size is smaller than concrete cover thickness in order to successfully capture the effect of cover cracking. Finally, a parametric study using difˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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Fig. 23. Three dimensional finite element model of the confined concrete column (Sheikh and Uzumeri, 1980).
Sheikh and Uzumeri (1980) - Column 2A1-1
4000
fc = 31.88 MPa
P(kN) 3500
3000
2500
2000 1500
1000
Analysis
Core concrete
Concrete cover
Experiment
500
0 0
0.001
0.002
0.003
0.004
0.005
0.006
Fig. 24. Load–displacement response, axial stress distribution and cover cracking of the reinforced concrete column.
ferent transverse reinforcement arrangements of increasing complexity (Fig. 25) clearly shows that the present constitutive model successfully captures the expected increase in strength and deformation capacity (reducing slope of the post-peak response curve) due to the passive confinement effect. ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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fc = 27.97 MPa 6000
P(kN) 5000
C
*Arrangement C
4000
3000
2000
Arrangement B
Longitudinal reinforcement 1000
Plain concrete
0 0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Sheikh and Uzumeri (1980) - Column 2C5-17 *
Arrangement A
Fig. 25. Increase in strength and deformation capacity of the reinforced concrete column, for increasing complexity of transverse reinforcement arrangements.
4. Conclusions A combined material model for concrete was presented, including an orthotropic smeared crack model for concrete cracking based on the Rankine failure criterion and a plasticity model for concrete crushing based on the Mene´trey–Willam criterion. The hardening/softening parameter for the plasticity model was related to the plastic volumetric strain, interacting with a nonlinear plastic potential function. Both models are formulated separately and a combination algorithm is developed to iteratively determine the separation of the strain increment into the fracturing and plastic part. The behavior of this algorithm was verified against several loading histories and reasonable correlation with experimental results was generally observed. The suggested model was successfully inteˇ ervenka et al., 2007) and was applied grated in the ATENA finite element software (C on the analysis of reinforced concrete beams under flexure and confined reinforced concrete columns under concentric compression. The comparison with experimental results was reasonable and it is therefore applicable for further applications on finite element analysis of plain and reinforced concrete structures. The proposed model could be further improved in the following areas: It is documented in the work by Collins (1978) that the compressive strength of cracked concrete should be reduced and should depend on the crack opening in other directions. This assumption may improve the accuracy in the prediction of the shear strength of reinforced concrete beams. Another important aspect is the shear behavior after cracking. The current model intuitively relates the shear stiffness of the cracked material to the normal crack opening law. There are experimental results in the literature, for instance Walraven (1980), which could be used to improve the shear behavior of the model. Finally, the plasticity part of the model could be further enhanced to include a cap ˇ ervenka, J., Papanikolaou, V.K., Three dimensional combined Please cite this article in press as: C ..., Int. J. Plasticity (2008), doi:10.1016/j.ijplas.2008.01.004
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surface for hydrostatic compression and handle the initial plastic compaction of concrete that is observed experimentally. Acknowledgements The contribution of the first author was supported by the research Grant number 1ET409870411 from the Czech Academy of Sciences. The financial support is greatly appreciated. The contribution of the second author was performed within the framework of the research project ‘‘ASProGe: Seismic Protection of Bridges”, funded by the General Secretariat of Research and Technology (GGET) of Greece. References Bazˇant, Z.P., Oh, B.H., 1983. Crack band theory for fracture of concrete. Materials and Structures, RILEM 16 (3), 155–177. Bazˇant, Z.P., Pijaudier-Cabot, G., 1987. Nonlocal continuum damage, localization instability and convergence. Journal of Applied Mechanics, ASME 55 (2), 287–293. Bazˇant, Z.P., Caner, F.C., Carol, I., Adley, M.D., Akers, S.A., 2000. Microplane model M4 for concrete: I Formulation with work-conjugate deviatoric stress. Journal of Engineering Mechanics, ASCE 126 (9), 944– 961. Bielger, M.W., Mehrabadi, M.M., 1995. An energy-based constitutive model for anisotropic solids subject to damage. Mechanics of Materials 19 (2–3), 151–164. Candappa, D.C., Sanjayan, J.G., Setunge, S., 2001. Complete triaxial stress–strain curves of high-strength concrete. Journal of Materials in Civil Engineering, ASCE 13 (3), 209–215. Carol, I., Rizzi, E., Willam, K., 1994. A unified theory of elastic degradation and damage based on a loading surface. International Journal of Solids and Structures 31 (30), 2835–2865. ˇ ervenka, V., Gerstle, K., 1971. Inelastic analysis of reinforced concrete panels Part I: Theory. Publication C I.A.B.S.E. 31 (11), 32–45. ˇ ervenka, V., Pukl, R., Ozbolt, J., Eligehausen, R., 1995. Mesh sensitivity effects in smeared finite element C analysis of concrete structures. In: Proceedings of the 2nd International Conference on Fracture Mechanics of Concrete Structures – FraMCoS 2, 1995, pp. 1387–1396. ˇ ervenka, J., C ˇ ervenka, V., Eligehausen, R., 1998. Fracture–plastic material model for concrete. Application to C analysis of powder actuated anchors. In: Mihashi, H., Rokugo, K. (Eds.), Proceedings of the 3rd International Conference on Fracture Mechanics of Concrete Structures – FraMCoS 3, Gifu, Japan, vol. 2. Aedificatio Publishers, Freiburg, Germany, pp. 1107–1116. ˇ ervenka, V., Jendele, L., C ˇ ervenka, J., 2007. ATENA Program Documentation. Part 1: Theory. C ˇ ervenka C Consulting, Prague, Czech Republic. Chiarelli, A.S., Shao, J.F., Hoteit, N., 2003. Modeling of elastoplastic damage behavior of a claystone. International Journal of Plasticity 19 (1), 23–45. Christoffersen, J., Mehrabadi, M.M., Nemat-Nasser, S., 1981. A micromechanical description of granular material behavior. Journal of Applied Mechanics, ASME 48 (2), 339–344. Cicekli, U., Voyiadjis, G.Z., Abu Al-Rub, R.K., 2007. A plasticity and anisotropic damage model for plain concrete. International Journal of Plasticity 23 (10–11), 1874–1900. Collins, M.P., 1978. Towards a rational theory for RC members in shear. Journal of the Structural Division, ASCE 104 (4), 649–666. Contrafatto, L., Cuomo, M., 2006. A framework of elastic–plastic damaging model for concrete under multiaxial stress states. International Journal of Plasticity 22 (12), 2272–2300. Dahl, K.K.B., 1992. A constitutive model for normal and high-strength concrete. ABK Report No. R287, Department of Structural Engineering, Technical University of Denmark. De Borst, R., 1986. Non-linear analysis of frictional materials. Ph.D. Thesis, Delft University of Technology, The Netherlands. De Borst, R., Mu¨hlhaus, H.B., 1992. Gradient dependant plasticity: formulation and algorithmic aspects. International Journal for Numerical Methods in Engineering 35 (3), 521–539.
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