A Continuum/Discontinuum Micro Plane Damage Model for Concrete

Transcription

1 A Continuum/Discontinuum icro Plane Damage odel for Concrete Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th Introduction S. A. Sadrnejad 1 and. abibzadeh 2 Deartment of Civil Engineering, K.N. Toosi University of Technology Tehran, Iran 1 sadrnejad@srttu.edu, sadrnejad@hotmail.com 2 mojtabalabibzadeh@yahoo.com Abstract: Analysis and rediction of structural resonse to static or dynamic loading requires rediction of concrete resonse tovariable load histories. The constitutive equations for the mechanical behavior of concrete caable of seeing damage effects or crack growth rocedure under loading/unloading/reloading was develoed uon micro-lane framework. The roosed damage formulation has been built on the basis of five fundamental tyes of stress/strain combinations, which essentially may occur on any of micro-lanes. odel verification under different loading/unloading/reloading stress/strain aths has been examined. The roosed model is caable of resenting re-failure history of stress/strain rogress on different redefined samling lanes through material. any of mechanical behavior asects haen during lasticity such as induced anisotroy, rotation of rincial stress/strain axes, localization of stress/strain, and even failure mechanism are redicted uon a simle rational way and can be resented. The fundamental characteristics of concrete behavior are established through exerimental testing of lain concrete secimens subjected to secific, relatively simle load histories. Continuum mechanics rovides a framework for develoing an analytical model that describe these fundamental characteristics. Exerimental data rovide additional information for refinement and calibration of the analytical model. The continuous models in turn consist of two large grous: macroscoic models in the context of damage and lasticity theory or combination of both and meso-scoic models such as micro-lane or multi-laminate models. The macroscoic models concern with the definition of relation between stress and strain tensors (structural scale) and the meso-scoic models deals with the same object but in the different way. The latter cature this goal by assigning of the relation between the stress and strain comonents of the different lanes with rescribed orientations called micro or multi lanes. Finally the microscoic models concern with the discrete article models consisting of convex olygons that are able to withstand a limited cohesion (granular scale). A descrition of contacts of articles as well as a bond formulation between them could lead to forces induced by article movements. These forces are inserted into the equations of motion, which are solved numerically based on the discrete element methodology. In this research, we ay our attention for the micro or multi lane models, which nowadays are develoed as owerful tools for numerical simulation of the geo and geolike materials. Nevertheless, these models have been formulated and used by many researchers in the recent years; it seems to be necessary to review these models in the new deeer mathematical way to revent any mistakes, which unfortunately have been made by the users of these models. 1.1From sli lanes to micro lanes The basic idea, namely that of the constitutive material behavior as a relation between strain and stress tensors can be 296 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

2 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 "assembled" from the behavior of material on the lanes with different orientations within the material such as sli lanes, micro cracks, article contacts, etc., might be traced back to the failure enveloes of ohr (1900) and the "sli theory of lasticity" of G.I. Taylor (1938) who was the first that imlemented this mentioned theory for modeling the behavior of olycrystalline metals. Taylor's idea was formulated in detail by Batdorf and Budiansky (1949). This theory was soon recognized as the most realistic constitutive model for lastic-hardening metals. It was refined in a number of subsequent works (e.g. in and Ito 1965, 1966, Kröner 1961, Budianski and Wu 1962, Hill, 1965, 1966, Rice, 1970). It was used in arguments about the hysical origin of strain hardening, and was shown to allow easy modeling of anisotroy as well as the vertex effects for loading increment to the side of a radial ath in stress sace. All the formulations considered that only the inelastic shear strains (`slis'), with no inelastic normal strain, were taking lace on what is now called the 'micro-lanes'. The theory was also adated to anisotroic rocks and soils under the name "multi laminate model" (Zienkiewicz and Pande 1977, Pande and Sharma 1981, 1982; Pande and Xiong 1982). It is interesting to note that in these works there is a common assumtion that the lanes of lastic sli in the material (in those studies called the `sli-lanes' and here in this article called the `micro-lanes') to be constrained statically to the stress (`macro-stress') tensor (i.e., the stress vector on each 'micro-lane' was the rojection of). The elastic strain was not included on the sli lanes but was added to the inelastic strain tensor on the continuum level (macro-level). The static constraint formulation was extensively used under the name of sli theory for metals or multilaminate theory for anisotroic rocks until the first alication of this theory by Bazant and Gambarova in 1984 and Bazant in 1984, for continuum damage mechanics and cohesive-frictional materials, which for the first time its name changed from sli theory or multi-laminate theory to micro-lane theory. Bazant and coworkers have been interested in this domain of research and tried to simulate the strain-softening of geomaterials with such a theory which was not of interest in the aforementioned studies. After a short time, they concluded that under the assumtion of the static constraint, a strain-softening constitutive law for the micro-lane makes the material unstable even if is rescribed. They suggested that it could be referable if in relace of static constraint, the kinetic constraint is used. In the kinetic constraint aroach, the strain tensor instead of stress tensor is rojected on the lanes. 2.Constraint Aroach: Equilibrium and Comatibility As it is mentioned in the revious section, before 1984, the early multi-lane models (called sli-lanes theory or multi-laminate models) develoed based on the static constraint formulation. Furthermore it was said that after 1984, Bazant and Gambarova and also Bazant and his assistants informed that to revent instability of numerical simulation of ost-eak behavior of cohesive-frictional materials it is necessary to use kinetic constraint formulation instead of static one. In this research, we will first focus on the suerimosition method, which used in the static constraint formulation in the ast and it will be shown that this formulation was not correct and then the reason of shortcomings of this aroach will be argue. Before 1984, in all the multi-lane models International Journal of Civil Engineerng. Vol. 4, No. 4, December

3 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 such as sli-lanes or multi-laminate models, first the macro-stress tensor was rojected on the micro-lanes and then by introducing onlane constitutive laws, the micro-strain comonents were calculated and finally the macro-strain tensor was identified by suerimosition of on-lane micro-strain comonents uon any of samling lane transformation matrix obtained through direction cosines of samling oints on the surface of a unit shere: x, y, zd 4 W f x, y z f, (1) Now consider the stress tensor in the macrolevel state in the center of the unit shere as always is used in the micro-lane models. Then, we are going to roject this tensor on the lanes, which are tangent on the surface of the shere in the rescribed oints. The number and osition of these oints are determined deending on the numerical integration formulation which elected for doing integration of an arbitrary function over the surface of the unit shere. It is worth noting that the origin of the initiation and roagation of the all multi-lane models including micro-lane or multi-laminate models are as this mathematical numerical formulation used for integration. Here, we use a recise formulation of 26 integration oints for this job. In the table 1, direction cosines and weights of the integration oints and in figure 1, their ositions on the surface of the unit shere are shown. If we roject the stress tensor on the surface of the shere, then we have: Fig.1 Position of integration oints on the unit shere surface Table 1 definition of micro-lanes N N, N,, n n i mi n j m jni / l n l n / 2 i j j j i 2 (2) Fig.2 Projection of stress tensor on the surface of unit shere 298 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

4 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 In which n i, i=1,2,3 are the direction cosines of the unit vector normal to the lane and m i, l i, i=1,2,3 are the direction cosines of two orthogonal unit vectors tangential to the lane. For convenience of calculations, one of the unit vectors tangential to the lane is considered to be horizontal (arallel to x-y lane). For instance, the rojection of the stress tensor on the lane number 1, results in: N x x y y z (3) Now, we try to obtain the original stress tensor from its rojections on the microlanes. To do so, it is first necessary to transfer every stress vector on the microlane from local coordinate to the global coordinate system and then we can add them u according to their weightings. Obviously, the result must be equal to numerical integration of the on-lane stress tensor. This numerical integration is a very crucial ste in the construction of any micro-lane model. To transfer every micro-stress vector to the macro state the following transition matrix can be written: T N N N N N N (4) Subscrit denotes any secified microlane. So we can write: 6 1 x y 2 z zx zx zx ˆ : N T. T. : N n1 m1 l1 n m l (5) (6) For examle, by transforming the microstress vector of the lane 1 to the macro level, we reach to the following six comonents vector: (7) Summing u the transformed six comonent vectors according of their weighting functions: (8) Subscrit denotes any secified microlane. As a general rule for the numerical integration of an arbitrary function f(x,y,z) over the surface of unit shere, we can use the following 26 samling oint equations as follows: (9) Comaring equation (8) and (9) we can write: n3 x m3 l 3 xz 2 x xz 2 y 1 2 z zy xz ˆ1 6 x y 2 xz y z 2 xz x z 2 xz 26 P W ˆ f x y z y xz n1 n2 n z 3 26 x, y, zd 4 W f x, y, z 1 zx T International Journal of Civil Engineerng. Vol. 4, No. 4, December

5 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Tσ. : Nd σˆ d 26 4 W ˆ x y T zx (10) So, to obtain the original stress tensor we can write the following equation: 3 4 (11) (12) This comarison shows that the remultilier in equation (12) and (1) for suerimosition is not the same so it is not correct. Furthermore, it is worth noting that in the static constraint aroach, the equilibrium of the forces in a oint are satisfied automatically because of the rojection of the stress tensor on the lanes, but the comatibility condition of strain tensor is met only in articular cases. In other words, the micro-strain comonents acting on the lanes may not be always as the rojection of the strain tensor, because the way of the suerimosition of micro-strain comonents which are used in the static constraint aroach (relation (1)) does not guarantee to be the same as the summation of the rojections of macro-strain tensor obtained on every lane. 2.1Kinetic constraint aroach: equilibrium no, comatibility yes! In 1984, Bazant and Gambarova suggested N. N W N N W. N N z... d. that instead of rojecting on-lane stress tensor, the strain tensor must be rojected. Considering the revious argument, in this aroach, static constraint has been relaced by kinetic constraint, however, the roblem is still not solved because the comatibility is satisfied but equilibrium condition is not met. Although Bazant and his assistants tried to enforce the static equilibrium by alication of the rincile of virtual work; at the end they realized that the micro-stress comonents would not equal to the rojection of macro-stress tensor. However, we note that the alication of rincile of virtual work is equivalent to establish the static equilibrium that latter itself equal to such a condition that micro-stress comonents on each lane are as the rojection of macro-stress tensor. In fact, on the other words, the static equilibrium condition is not satisfied if and only if the micro-stress comonents that comuted on the micro-lanes are as the rojections of the macro-stress tensor. So, regarding to the above rational argument and according to the investigations done by the authors, it is concluded that the formulation which Bazant and coworkers named it as rincile of virtual work, basically is the only valid suerimosition formula which was formerly derived by the authors and could not be the same as the alication of usual virtual work (see relation13, which derived by Bazant and comare it with relation 12). Nm s d N N 3 (13) Furthermore it is worth noting that, equation (12) that has been derived based on the 26 integration oint technique, is comletely accurate but the equation (13) is fairly accurate. w s, s 300 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

6 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th Double Constraint Aroach To satisfy both of static equilibrium and comatibility conditions, we have considered a new method as rojecting the stress tensor on the micro-lanes as described earlier in this article. Then, we derived the strain tensor in terms of the stress tensor based on a wellcaable constitutive relation in an ordinary three-dimensional coordinate system. In the second case, the derived strain tensor was rojected or transformed on the microlanes. So, in this stage, by comaring the comonents of stress and strain on the microlanes we must be able to define the equal microscoic constitutive relations in such a way that both of the stress and strain comonents on each micro-lane are as the rojections of the corresonding stress and strain tensors. This situation, in fact, is the double constraint formulation in which the equilibrium of forces and comatibility of dislacements in every integration oints are satisfied one by one. 3. Detachment of Behavior to Devitoric and Volumetric Parts In order to attain to the double constraint asect, after analogy of the rojections of stress and strain tensors on the micro-lanes obtained in the manner that was exlained in the revious section, it was certain that it is necessary to searate the behavior of material into two distinct arts as deviatoric and volumetric. So if we discrete the strain tensor as the volumetric and deviatory arts firstly and then roject each of them on the microlanes searately, we may try to obtain the deviatoric art of the modules matrix from the behaviors which are taking lace on the micro-lanes and the volumetric one which is not affected by the direction characteristics and essentially is isotroic, obtained in the ordinary coordinate system and summed u to the deviatoric art at the end of each ste of loading. Therefore, we can write: 3 E δ δkl Dkl N Nkl kl kl dω 4π Ω 1 ν 3 3 E δkl δ (14) 12ν 3 4.Anistroy Damage odel Total deviatory art of constitutive matrices is comuted from suerosition of its counterarts on the micro-lanes that such counterarts in turn, are calculated based on the damages occurred on each lane deending on its secific loading conditions. This damage is evaluated according to the five searate damage functions; each of them belongs to the articular loading states. This five loading conditions are as: -hydrostatic comression, -hydrostatic extension, -ure shear, -shear + comression, -shear + extension. On each micro-lane at each time of loading history, there exists one secific loading situation that it may be in one of the five mentioned basic loading conditions. For every five mood, a secific damage function according to the authoritative laboratory test results available in the literature is assigned. Then, for each state of on lane loading, one of the five introduced damage functions will be comuted with resect to the history of micro-stress and strain comonents. 4.1 odel Parameters In this formulation we consider just two basic material arameters for ease as elasticity and Poisson's coefficients. International Journal of Civil Engineerng. Vol. 4, No. 4, December

7 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.3 Positions of the micro-lanes in a cubic 4.2Correlation Studies To establish the validity of the roosed concrete material model, correlation studies of analytical results with exerimental evidence from the stress-strain resonse of concrete secimens under different loading conditions are resented in the following. Uniaxial Comression (UC) test As can be seen in the figure 5, there is a good agreement between the results that were fulfilled by the roosed model and exerimental evidences. The material arameters used in the above analysis are as: E=25000Pa, v=0.20. In the figure 6, the volumetric changes of the concrete secimen under uniaxial comressive loading have been comared with the exerimental observations exerienced by Kufer and his co-workers in Obviously, there exists an excellent coincidence between analytical and laboratory data. To show more confidence on the caability of the micro-lanes during uniaxial comression test, in the figure 7, the variation of micro-stress normal and tangential comonent values are reresented verses the total axial comressive stress. As can be seen from figure 7, during the alication of the uniaxial comressive load on the x-axis, the micro-lane number 11 (see figure 1) is under just the comressive stress whereas the micro- lanes number 9,10,12,13 which geometrically are located normal to the load direction on the unit shere are only under the tensile stress. Comressive stress accomanied with shear affect the other remaining lanes. It is interesting to note that during increase of the uniaxial comressive load, the comressive and shear stress comonents acting on the micro-lanes number 1 to 8 increase together with more rise of shear stress at first, but near to the eak stress ( f c t ) the comressive stress decreases suddenly. Figure 8 shows the growth of the damage function values of different micro-lanes during uniaxial comression test of concrete obtained with the roosed model. As it can be well observed from this figure, damage evolutes faster on the micro-lanes number 9,10,12,13 on the unit shere than the other lanes. On lane no. 11; there exists only a normal comressive load (mode I) by which no damages could be occurred on it. On the micro-lanes number 5,6,7,8 there is a shear combined with the normal comressive load (mode IV) causes damages less than the micro-lanes 1,2,3,4 on which there exists a same mode of loading (mode III). This is because of the fact that on the micro-lanes number 1, 2, 3 and 4 the magnitude of the comressive stress comonent is less than the same comonent value on the micro-lanes number 5 to 8 (see figure 7), so damage growths faster. Finally, on the micro-lanes number 9,10,12,13 there exist only normal tension loading (mode II), causes the damage growths faster than the all other lanes. This is introduced from roosed model that in the uniaxial comression test, the damages or cracks can be aeared first on the micro- 302 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

8 Star Read the material arameters and the number of maximum iteration Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Iterates on the number of the micro-lanes Calculates the volumetric art of the undamaged modules matrix Read the direction cosines and the weights of the micro-lanes according to table 1 Calculates the transortation matrixes of micro-lanes Calculates the transose of the transition matrixes of micro-lanes Reads the strain tensor Calculates the deviatory strain tensor Projects the deviatory strain tensor on the micro-lane Comutes the deviatory damage function of the micro-lane Comutes the deviatory damage modules matrix of the micro-lane Transorts the deviatory damage modules matrix of the micro-lane to the ordinary Cartesian coordinate system Assembles the deviatory damage modules matrix of the micro-lanes and calculates the deviatory damage modules matrix art of the total modules matrix in the ordinary Cartesian coordinate system Iterates on the number of maximum inut data Comutes the volumetric damage function Comutes the volumetric art of the total modules matrix Comutes the total modules matrix Calculates the stress tensor End Fig.4 Oeration sequence of the roosed micro-lane damage model develoed in the VISUA FORTRAN comuter language International Journal of Civil Engineerng. Vol. 4, No. 4, December

9 Axial stress (Pa) f c 27Pa Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Kufer.et. al., ateral strain (mm/mm) Axial comressive stress (Pa) Kufer.et. al., 1969 Axial strain (mm/mm) uniaxial comression test of concrete obtained with roosed micro-lane damage model Fig.5 Axial and lateral strains versus axial stress in uniaxial comression test of concrete obtained with roosed microlane damage model f c 27Pa 10 5 Kufer.et. al., Contraction Volumetric strain (mm/mm) Dilation Fig.6 Volumetric behavior of concrete under uniaxial comressive loading 304 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

10 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.7 Variation of micro-stress comonent values during uniaxial comression test Fig.8 Comarison of the damage evolution functions on the various micro-lanes during the axial comressive loading International Journal of Civil Engineerng. Vol. 4, No. 4, December

11 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.9 Tyical failure atterns of cylinders in uniaxial comression test with a) low frictional b) high frictional restraint lanes number 9,10,12,13 and then on the micro-lanes number 1, 2, 3 and 4. This can be so observed in the real situation of the laboratory on the cylindrical concrete secimen. If there is no friction restraint between the surfaces of the loading to/bottom lates and the secimen, the cracks will be aeared differently on the ositions of the micro-lanes number 9,10,12,13 of the roosed model. Else if the damages on the micro-lanes number 1, 2, 3 and 4 will be greater and cracks will be initiated first on these lanes. This henomenon is deicted in the figure 9. Conventional Triaxial Comression (CTC) test In this test, at first the hydrostatic ressure is alied to the secimen to a certain level and then the axial comression is increased while the lateral or confining ressure held constant. So in this test, u to the certain level of hydrostatic comression there must be no any shear forces on the micro-lanes. This can be seen in figure 10 that shows the evolution of the micro-stress comonents on the different micro-lanes during CTC test. In figure 11, the axial stress-strain curves of cylindrical concrete secimen under two different Uniaxial Comression (UC) test and Conventional Triaxial Comression (CTC) test obtained with roosed microlane damage model have been comared. As it is observed from this result, the alication of initial confining ressure of about %30 increases comression strength by( 0.3Gf c t ), although, it can be imroved aroximately u to ( 1.4Gf c t ). As a result, the effect of lateral confining ressures on the comressive cylindrical strength of concrete secimens simulated by roosed model has been comared with exerimental data of Ansari and i (1998) in figure 12. Conventional Triaxial Extension (CTE) test To clarify the integrity of the roosed model in various loading conditions, this test is also considered. In this test, at first, the hydrostatic ressure is alied to the secimen to a certain level, but after that, the axial comression is decreased while the lateral ressure is held constant. The stress comonent ermutation during the CTE test on the secimen is shown in figure 13. As it can be seen in figure 13, at the beginning stages of the test, the shear stress comonents on all of micro-lanes are zero and normal comonents are increased linearly. During decrease of axial comression and creation of shear forces consequently, the shear stress comonents are aeared on the inclined micro-lanes number 1 to 8 while at the same time the normal comonents are develoed on the whole lanes (excetion of lane 11 on which the normal stress is ket constant according to the test results) and got to its maximum values when the eak axial stress is reached. Therefore, the redicted behavior of micro-lanes under the CTE test is erfectly logical. 306 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

12 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.10 Variation of micro-stress comonent values during Conventional Triaxial Comression (CTC) test Fig.11 Comarison of axial stress-strain curves of concrete in UC and CTC tests Fig.12 Different triaxial comression strengths f cc obtained with resect to different lateral confining ressures f t International Journal of Civil Engineerng. Vol. 4, No. 4, December

13 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.13 Variation of micro-stress comonent values during Conventional Triaxial Extension (CTE) test Fig.14 Behavior of cylindrical concrete secimen under Conventional Triaxial Extension (CTE) test obtained with roosed micro-lane damage model Fig.15 Behavior of cylindrical concrete secimen under uniaxial tension test obtained with roosed micro-lane damage model 308 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

14 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.16 Variation of micro-stress comonent values during Uniaxial Tension (UT) test Uniaxial Tension (UT) test The stress-strain resonse of the concrete cylindrical secimen under axial tension load is deicted in figure 15. Revenue mechanism of micro-lanes under the action of UT test is reresented in figure 16. As it is exected, the damage on the inclined lanes 1 to 8 is due to comromise of tensile and shears action. If we comare this icture with figure 7, the combination of tension and shear forces will roduce damages much more quickly than comression and shear forces. Hydrostatic Tension (HT) test In figure 17 the roosed model rediction under HT test is deicted. As it is waiting for, the load caacity of the samle under the hydrostatic tension is greater than the same value in the uniaxial tension test because in the uniaxial tension test the cause of the damage is the comromise of tension and shear stress while in the hydrostatic tension test, ure tensile stress acting on the lanes. The rediction of the roosed micro-lanes International Journal of Civil Engineerng. Vol. 4, No. 4, December

15 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 Fig.17 Behavior of cylindrical concrete secimen under Hydrostatic Tension test obtained with roosed microlane damage model Fig.18 Variation of micro-stress comonent values acting on the micro-lanes number 1 to 13 during Hydrostatic Tension (HT) test Fig.20 Variation of micro-stress comonent values acting on the micro-lanes number 1 to 13 during Hydrostatic Comression (HC) test Fig.21 cyclic comression test simulation Fig.19 Behavior of cylindrical concrete secimen under Hydrostatic Comression test obtained with roosed micro-lane damage model Fig.22 comlete cyclic test simulation 310 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

16 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 for the HT test is shown in the figure 18. As it can be observed the behavior of all the micro-lanes under this loading is the same. The reson is that in the hydrostatic loading, the same condition of stress distributions are imosed around the hysical oint and therefore the anisotroy could not to be aeared. Furthermore as it is anticiated, there is no any shear stress on the lanes. Hydrostatic Comression (HC) test The exected resonse of concrete cylindrical secimen under hydrostatic comressive loading, associated with the increase of bearing caacity unlimitedly. Simulation of this resonse under the HC test is resented in figure 19, 20. As in the case of hydrostatic tension test, the behavior of the whole lanes under the HC test is the same. Cyclic loading Generally, the most damage models fail to reroduce the irreversible strains and the sloes of the curve in unloading and reloading regions. To overcome this roblem, often the lasticity and damage models are combined. In figure 21, the redicted resonse of the model under cyclic comression test is comared with the exerimental results of B.P. Sinha, K.H. Gerstle and.g. Tulin (1964). Obviously, there is a good agreement between the analytical and exerimental data. Also, redicted behavior of concrete under comlete cyclic loading is shown in figure Conclusion A constitutive damage model for the mechanical behavior of concrete under any arbitrary loading develoed using the comosition of a new theoretical micro-lane framework. The roosed model is a micro-lanes based, uon a certain micro-lane based stress/strains, satisfying both equilibrium and comatibility conditions. This characteristic of the model is called double constraint. A new damage formulation has been emloyed into the micro-lane model. This damage formulation has been built on the basis of five fundamental force conditions that essentially can be occurred on each micro-lane. Consequently, any arbitrary change of six strain/stress comonents led to a combination of five introduced on lane coditions. Therefore, the roosed model is caable of redicting the concrete behavior under any arbitrary strain/stress ath. These five force conditions are as: - hydrostatic comression, - hydrostatic extension, - ure shear, - shear + comression, - shear + extension. The five damage evolution are function of equivalent strain were formulated for any of the five stated conditions. The equivalent strain for the two first conditions are defined as limitation in volumetric strain and for the others is the suerimosed of rojections of deviatoric strain tensor on the corresonding micro-lane. These damage functions are constructed with resect to the exerimental evidences on the concrete secimens under comressive and tensile loading conditions. The roosed model has excellent features such as refailure strain distribution inside material led to final failure mechanism, caability of seeing induced/inherent anisotroy and also any fabric effects on material behavior. However the basis of its International Journal of Civil Engineerng. Vol. 4, No. 4, December

17 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 formulation is simle, logical and has some hysical insights that make it convenient to erceive. References [1] [2] [3] [4] [5] Batdorf, S.B., Budiansky, B. "A mathematical theory of lasticity based on the concet of sli.", Technical Note 1871, National Advisory Committee for Aeronautics., Ramm, E., G. A. D'Addetta, and. eukart."from microscoic to macroscoic modeling of geomaterials". Euroean Congress on Comutational ethods in Alied Sciences and Engineering, Bazant, Z.P., P.G. Gambarova. "Crack shear in concrete: Crack band micro lane model." J. Struct. Eng., ASCE, 110, , Bazant, Z., B.Oh. "icro lane model for rogressive fracture of concrete and rock." J. E. ech., 111, , Bazant, Z., P. Prat. "icro lane model for brittle lastic material: Part I & II. "J. E. ech., 114, , Carol, I., Z. Bazant. "Damage and lasticity in micro lane theory." Int. J. Solids & Structures., 34, [8] [9] [10] [11] [12] "A thermodynamically consistent aroach to micro lane theory. Part I: Free energy and consistent micro lane stresses." Int. J. Solids & Structures, 38, , Kuhl, E., E. Ramm and R. de Borst. "An anisotroic gradient damage model for quasi-brittle materials.", Com. eth. Al. ech. Eng., 183, , Kuhl, E. and E. Ramm. "icro lane modeling of cohesive frictional materials. ", Eur. J. ech." A/Solids, 19, S121-S143, Simo, J. and J. Ju. "Strain and stress based continuum damage models: Part I-Formulation, Part II-Comutational asects.", Int. J. Solids and Structures 23, , Bazant, Z. P., Adley,. D., Carol, I., Jirasek,., Akers, S.A., Rohani, B., Cargile, J.D., Caner, F.C. "arge-strain generalization of micro lane constitutive model for concrete and alication." ASCE, J. Engrg. ech. 126(9), , 2000a. Bazant, Z. P., Caner, F. C., Carol, I., Adley,.D., Akers, S.A. "icro lane model 4 for concrete: I. Formulation with work-conjugate deviatoric stress. "ASCE, J. Engrg. ech. 126(9), , 2000b. [6] [7] Carol, I., Z. P. Bazant and P. Prat. "New exlicit microlane model for concrete: Theoritical asects and numerical imlementation." Int. J. Solids & Structures, 29, , Carol, I.,. Jirasek and Z. P. Bazant. [13] [14] Bazant, Z. P., Caner, F. C. "icro lane 4 for concrete: II. Algorithm and calibration. ASCE, J. Engrg. ech. 126, , Ozbolt, J., i, Y., Kozar, I. "icro lane model for concrete with relaxed 312 International Journal of Civil Engineerng. Vol. 4, No. 4, December 2006

18 Downloaded from ce.iust.ac.ir at 17:17 IRST on Sunday January 6th 2019 [15] [16] [17] kinematic constraint.", Int. J. Solids Struct. 38, , Paudier-Cabot, G., Z. P. Bazant. "Non local damage theory.", J. Engrg. ech. ASCE, 113, , azars, J. "A descrition of micro and macro scale damage of concrete structures.", J. Engrg. Fracture ech., 25, , eschke, G., ackner, R., ang, H. "An anisotroic elastolastic-damage [18] [19] model for lane concrete.", Int. J. Numerical ethods in Engrg., 42, , Yazdani, S. and Schreyer, H.. "An anisotroic damage model with dilatation for concrete.", echanics of aterials 7, , ubarda, V. A. and Krajcinovic, D."Damage tensor and the crack density distribution.", Int. J. Solids Structures, Vol. 30, No. 20, , International Journal of Civil Engineerng. Vol. 4, No. 4, December