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1、Application of thermodynamics-based rate-dependent constitutive models of concrete in the seismic analysis of concrete dams Abstract: This paper discusses the seismic analysis of concrete dams with consideration of material nonlinearity. Based on a consistent rate-dependent model and two thermodynam

2、ics-based models, two thermodynamics-based rate-dependent constitutive models were developed with consideration of the influence of the strain rate. They can describe the dynamic behavior of concrete and be applied to nonlinear seismic analysis of concrete dams taking into account the rate sensitivi

3、ty of concrete. With the two models, a nonlinear analysis of the seismic response of the Koyna Gravity Dam and the Dagangshan Arch Dam was conducted. The results were compared with those of a linear elastic model and two rate-independent thermodynamics-based constitutive models, and the influences o

4、f constitutive models and strain rate on the seismic response of concrete dams were discussed. It can be concluded from the analysis that, during seismic response, the tensile stress is the control stress in the design and seismic safety evaluation of concrete dams. In different models, the plastic

5、strain and plastic strain rate of concrete dams show a similar distribution. When the influence of the strain rate is considered, the maximum plastic strain and plastic strain rate decrease. Key words: concrete； constitutive model； rate dependency； concrete dam； nonlinearity； seismic analysis 1 Intr

6、oduction China has abundant water resources. As part of the national energy and water conservancy plan, a batch of 300 m-level high concrete arch dams are or will soon be under construction. Most of the dams are situated in regions of strong seismic activity. Their designed seismic accelerations rea

7、ch 0.2g-0.32g ( g = 9.81 m/s2 ), with a probability of exceedance of 2% over a 100-year period. The designed maximum acceleration of the Dagangshan Arch Dam even reaches 0.5575g. The influence of earthquakes should be considered and cautious arguments and investigations should be made about the seis

8、mic safety of the dams to ensure their safe operation and minimize damage from earthquakes. Great progress has been made in the field of technology for dynamic analysis of concrete dams during earthquakes. However, some issues need to be better understood, including the nonlinear constitutive model

9、and dynamic behavior of concrete, which are still based on elastic analysis (Lin and Chen 2001； IWHR 1997). A rational constitutive model of concrete is the foundation of further research on thenonlinear behavior of concrete. An appropriate constitutive model can reflect the mechanical characteristi

10、cs of the material in all the stages of deformation, including strain hardening and softening, strength reduction, and stiffness degradation, which determine the progressive damage 温州大学本科毕业设计 建筑与土木工程学院 1 of concrete. After several decades of development, the plastic theory describes these characteri

11、stics of concrete with a rather good theoretical basis. Nevertheless, there are too many hypotheses in the present plastic and viscoplastic models of concrete, including the Drucker postulate, associated or non-associated flow rules, and plastic potential functions, some of which have no clear physi

12、cal significance and others of which do not conform to the thermodynamic laws (Leng et al. 2008). For the nonlinear analysis of concrete dams, it has become important to construct a constitutive model with few hypotheses, which can conform to the energy principle and reflect the nonlinear behavior o

13、f concrete. In the meantime, concrete is a typical rate-sensitive material, whose strength, stiffness and ductility (or brittleness) are subject to a loading speed. Obviously, considerable deviation would be caused by the static mechanical parameters for seismic analysis. Identifying material charac

14、teristics of concrete under different strain rates and establishing proper dynamic constitutive models of concrete have become prerequisites for nonlinear dynamic analysis of arch dams. The descriptions of dynamic behavior of concrete in the Specifications for Seismic Design of Hydraulic Structures

15、(IWHR 1997) are very simple and need further supplement and improvement. Based on a consistent viscoplastic model, two thermodynamics-based consistent rate-dependent models were derived from two thermodynamics-based static models with consideration of the influence of the strain rate. They satisfy t

16、hermodynamic laws automatically and can authentically describe the dynamic behavior of concrete. With the two models, the seismic responses of the Koyna Gravity Dam and the Dagangshan Arch Dam were analyzed. The nonlinear dynamic behavior of concrete dams and the influence of the strain rate on the

17、dynamic behavior of concrete dams were discussed through comparison with the linear elastic model and the rate-independent models. 2 Introduction to thermodynamics-based rate-dependent constitutive models of concrete On the basis of the classical plastic theory, a consistent rate-dependent model tak

18、es into consideration the influence of the strain rate and maintains that stress remains on the yield surface in viscoplastic flow (Wang 1997). Therefore, the consistency condition is satisfied. The yield criteria with the strain rate effect can be expressed generally as ( , , ) 0f 0d （ 1） 温州大学本科毕业设

19、计 建筑与土木工程学院 2 where is the stress tensor, is an internal variable, is the rate of the internal variable, and d is the viscoplastic multiplier. Due to the rarity of multiaxial dynamic experiments of concrete at present, the multiaxialdynamic constitutive relationship cannot be established directly. I

20、t is assumed for simplicitys sake that the dynamic increase factor (DIF) of multiaxial strength is the same as that of uniaxial strength. Concrete shows completely different behavior under tension and under compression. Therefore, the internal variable is split into two parts, ck and tk , and behavi

21、ors under tension and under compression are described separately by means of these two internal variables. Under uniaxial compressive or tensile loading, we have ( , , )c c cf f k k , ( , , )t t tf f k k （ 2） and in a complex stress state, we have ()cckk ， ()ttkk （ 3） where ()c and ()t are the weigh

22、ting functions of compressive and tensile internal variables, respectively. The principles for determining ()c and ()t are as follows: for the loading process with dominant tensile stress states, ()t=1 and ()c=0 ； similarly, for the loading process with dominant compressive stress states, ()t=0 and

23、()c=1； for other loading conditions, 0 ()t1, 0 ()c1 , and ()t+ ()c =1. Finally, the yield criterion can be expressed as ( , , ) ( , , , , ) 0c t c tf k k f k k k k； (4) According to the theory of the consistent rate-dependent model, a consistency condition should be satisfied: f ： 0ctct ctctf f f fd

24、 d d d dkk kk (5) Two thermodynamics-based plastic constitutive models of concrete that have simple forms and can satisfy the energy laws naturally have been developed by Leng et al. (2008). Results from both models, obtained under static loading, agree with experimental results. The two models have been established, respectively, in Haigh-Westergaard stress space and principle stress space, the difference being that the Lode angle is neglected in the Haigh-Westergaard stress space model and considered in the principle stress space model. The two yield criteria can be expressed as