Modeling of crack nucleating damage of concrete

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ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 31 (2012) 282 – 287

Procedia Engineering www.elsevier.com/locate/procedia

International Conference on Advances in Computational Modeling and Simulation

Modeling of crack nucleating damage of concrete Kunxiong Zhoua, Lixiang Zhanga* a

Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming 650051, China

Abstract Based on the concept of renormalization group, a methodology of crack-nucleating cumulative damage of concrete is studied and three damage-nucleating models of stress-strain relations were obtained. The jump of multi-scaling and two-leveling phase changes of damage state is defined by a fractal length on a damage body. The damaging models established in this paper include influences of aggregate size, aggregate interface characteristics, crack nucleation, fractal dimension, etc. on concrete damage by using energy-equilibrium law in material under the state in the cracknucleating damage.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Kunming University of Science and Technology Keywords: Concrete; cumulative damage; crack nucleation; fatigue.

1. Introduction Concrete is a makeup material that is made of cement, sand, gravel, and water, etc. In from mechanical viewpoint, it can be described as a man-made material whose physical structure is discontinuous, nonhomogeneous, and irregular in micro or meso level due to interfaces of material phases. The mechanism of failure has been well unsolved so far. Generally, it is considered that the interfaces between the different scale components (material phase ) in concrete create lots of inherently weak joints and micro-, mesco-, and macro-cracks, as well as voids, or say defects, will be produced there. A concrete body contaminated by the defects informs a non-ordered system of cracks. The damage [1-3] gets up firstly at the weakly jointed places at which the crack nucleates and the damage evolution starts and diffuses with

* Corresponding author. Tel.: +86-0871-3303561; fax: +86-0871-3303561. E-mail address: [email protected]

1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.1025

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growth of the micro-cracks with loading [4]. Thus, the failure problem of concrete may be characterized by a multi scale and multi level cumulative damage mechanism. In the present study, we use methodology of the fractal geometry to describe the configuration of the damaged front and establish mathematical models of describing relation between stress and characteristics of the material phase interfaces in concrete. 2. Fractal length of damage nucleating cracks We assume an arbitrary damage-scaling level is Di , and a set of the activated micro-cracks in concrete is stated with a group,  i . The configuration of Di is assumed to be a fractal cell as shown in Fig.1, whose circumference is featured with a fractal line[5] around this cell. The nucleating damage, going with nucleation of the micro-cracks, starts firstly at a place in the circumference of the fractal cells when the strain or the stress exceeds the nucleating-damage threshold of concrete. Let the fractal cell be a damaged body consisting of 4 quadrants, and a micro-crack appears stochastically at arbitrary quadrant. Obviously, if the micro-cracks are simultaneously produced at 4 quadrants of the fractal round, the damaged body is in fully damaged state. Subsequently, if the micro-cracks are measured at the symmetrical quadrants, say quadrants 1 and 3 or quadrants 2 and 4, the damaged body is also considered to be in damaged state.

p B  2 p 2 (1  p) 2  0 undamaged

2

damaged

undamaged

Fig.1 Describing of damaged cell

Thus, using the concept of the K. G. Wilson's renormalization group methodology, a three magnification transformation of renormalization group  i is created as:



pi   i  2  pi   pi4  4 pi3 1  pi



(1)

where p i is the priori crack-nucleating probability, p i is the probability after twice renormalization magnification. A physical interpretation of this semi-group transformation is a twice-amplifying relationship of the measuring scale of the cracks. Therefore, the critical jump point of the damage phase change is equated as:





pc4  4 pc3 1  pc  pc  0

(2)

The solutions of this equation are pc = 0, 1, 0.768, and -0.434, respectively. It is clear that the solutions of 0 and 1 represent the zero and completely damaged states, respectively, minus number of the solution should be dropped out because of no physic meaning （ a probability does not appear in a minus

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number）. Thus, the probability of the damage phase change is pc = 0.768, and the expectation value of nucleating damage at the defined damaged body is stated as:







Ec  4 pc4  4 pc3 1  pc / pc  4 pc2

(3)

The value, substituting the critical value of 0.768, is Ec = 2.359. In fact, it is noted that Ec is 2.0 for the classical fracture mechanics. Assuming the measuring scale  is as much as n 1 times of the diameter of the fractal round under consideration, namely d  n , the fractal dimension of the damage group, D f , is expressed as:

Df 

ln E c ln E c  d ln n ln

(4)



The relationship between the fractal dimension and the measuring scale is shown in Fig. 2.

Fig. 2 Fractal versus measuring scale

It follows the Mandelbort's definition of fractal line, the circumference length, L f , of the fractal round can be stated as:

L f    L0  

1 D f

(5)

As an example, if the fractal cell is considered as a fractal round and the nucleating cracks appear at the quadrant 1 and 3 of the fractal round, the fractal length of the micro-crack after nucleating is as following: 1 1 D f Lc    L0   (6) 2

If L0   is defined as the circumference length of the regulation round, namely L0    n , the expectation length of the micro crack at the damaged body is written as: 1 2 D f Lc    E c n (7) 4 Thus, we get a formula that expresses the crack length in the measuring scale and the fractal dimension, namely: Lc    0.768n

3. Damage nucleating modeling of concrete

D f 1

d

2 D f

(8)

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Kunxiong Zhou and ZhangEngineering / Procedia Engineering 31 (2012) 282 – 287 Kunxiong Zhou et Lixiang al./ Procedia 00 (2011) 000–000

In case of the state of the crack-nucleating damage, the equilibrium condition of the unit-thicken concrete body is stated as: 2  4  0     f    C3   d    

2

 1  C3   4E 2 1 d  1   Lc 

0



(9)



where δ0 is the thickness of cement-grouting film between the defect (fractal round) and concrete, σf the failure strength of concrete, σ the stress, C3 the material constant to be determined by test,  energy containing in unit volume of concrete,  Poisson's ratio, and E initial elastic modulus. It can be seen that the right-hand side of the equation is defined by simulating Griffith's law. Introducing the foregoing definitions to the fractal round and the length of the micro-crack, equation (9) can be changed as: 2  4  0     f    C3   d    

2

2 D  E .1 f 1  C3   2 6.D472 1 2 D d  n f 1  2 d f 

0



(10)



If the problem under consideration is focused on a comparatively big scale granule diameter of concrete aggregate, namely only considering d   0 , the first term of the left hand side of equation (10) can be dropped out, and a crack-nucleating damage stress is yielded as:

  

 dc   C3 



0

4



1  C3 

d

2



2 D f

6.472E .1

 2n

D f 1

1  d 2

(11)

2 D f

The damage energy is introduced and divided into two parts. One is of a contribution of deteriorated damage, due to deterioration of the material characteristics by crack nucleation; and other is of releasing energy due to formation of the micro-cracks at the damaged area. To the former, Helmholtz's free-energy expression is employed, namely: 1  d    2  ii  jj    ii  jj   ij  ji  C1 Ddi  ij Ddj  kk  C 2 Ddi  ij  jk Ddk (12) 2 where  is the mass intensity of the material, εij the strain components,  ,  the Lame' constants, and C1, C2 the material parameters to be determined by test and to be related to the damaged extent. For the sake of simplification, the tension state of the material is only considered in this paper, and after a lengthily deducing, the deteriorated damage strain energy, ψd, is obtained as:



d 

2 2 E 1     1   1  1   1  2   2  1 



  2 C1  C 2  2C1 3 1   C12 4  11    11   11 2  3B1  B3  8 B1  B3   

(13)

and the releasing energy due to cracking is as:

c 

E 1   2 Lc

41   2 1  2 2

 2 2 1   1  

  C  C 2  2C1 2 1   C12 3  11  1     11 11  B1  B3  2 B1  B3 2  

2

(14)

where B1, B3 are the material parameters. If the hyperbolic damage front[6] is assumed to describe developing damage inside concrete body with loading, the following relationship is used as

B1  B3    112 Dd

Thus,

(15)

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d 

E 1    1  2 2  1  1   1  2   2  1 

c 

 2 E 1   2 n f d f 6.4721   2 1  2 2 D 1 2 D

 C1  C2  2C1  2 1   2 4  2  Dd  C1 Dd  11  3 8  

 2 2 1   1  

(16) 2

 2    C1  C 2  2C1 Dd2  1   C12 Dd4   11  2  

(17)

A substitution of equation (16) and/or (17) into equation (11) obtains the following damage constitute models describing concrete in tensile state. Model I, unique consideration of damage-nucleating energy (by substituting equation (17) only)

 dI 

 2 2 1   1  

    C1  C 2  2C1 Dd2  1   C12 Dd4   2   1   E ,    th 11 11 2 1     4 0 1   1  2  C3  1  C3   d  

(18)

Model II, non-consideration of damage-nucleating energy (by substituting equation (16) only)

 dII 

1  2 2 6.472 1    2  1   2 D f 1

 n

 C1  C 2  2C1  2 1    2 4   Dd  C1 Dd   3 8  

1    1  2 d 2

2 D f

  C3  4   

 0 1  C3  d 

4

E 11 ,  11   th

(19)

Model III, integrated damage (by substituting equations (16) and (17)) 2  dIII   dI2   dII

(20)

4. Discussions of parameters C1 and C2 To illustrate the procedure to determine the material parameters of C1 and C2, the fatigue tests of 32 concrete specimens of 70.7mm70.7mm212.1mm were done by a machine of MTS 810 TestStar, and 3parameter Weibull distribution is used to describe the probabilistic characteristics of the strains. Based on the definition of the P-Dd-ε curve of concrete[7], a regressive curve on Dd-ε, corresponding to the expectation probability pc = 0.5 (closely to the foregoing theoretical value of 0.768), is obtained as:

  803Dd 3  910Dd 2  501Dd  73

(21)

Clearly when Dd = 0, the threshold strain of the damage is  th  73 . In considering case of Dd = 0.2 , it is easy to get the relevant value of the strain from equation (21) as 143.2  . Thus, dynamic stress corresponding to this strain is approximately got as 2.7MPa by test constitutive relationship[8]. Taking  0 / d  0.001 , C3 = 0.2,   0.16 , E  2.1  104 MPa , d = 0.01m, n = 10, and Df =0.393, the relationships between C1 and C2 are obtained, from equations (18) and (19), and shown in Fig. 3, respectively, as: C12  4C1  5.97C2  133.4  0

(22)

C12  54.33C1  79.60C2  2802.98  0

(23)

Kunxiong Lixiang ZhangEngineering / Procedia Engineering 31 (2012) 282 – 287 KunxiongZhou Zhouand et al./ Procedia 00 (2011) 000–000

Fig. 3 C1 versus C2 in case of the above conditions

5. Conclusions In consideration of the special characteristics of concrete makeup, a combination of the mechanisms of both damage and micro-crack nucleation is a way to study crack-nucleating damage of the hydraulic concrete. From the theoretical viewpoints, the crack-nucleating damage models established in this paper are suitable. It is surely that a result closer to the practice damage and fracture in engineering can be well predicted by the viewpoints subjected, if the further work is done, especially in experimental and comparing studies. Acknowledgements The authors gratefully acknowledge the support from the National Natural Science Foundations of China (No. 50839003) and Yunnan (No. 2008GA027). References [1] L. M. Kachanov. Introduction of continuum damage mechanics. Martinus Nijhoff Publishers, Dordrecht, 1986 [2] R. Ilankamban and D. Krajcinovic, A constitutive theory for progressively deteriorating brittle solids. Int. J Solids Structures, 23(1987): 1521-34 [3] A. Dragon and Z. Mroz, A continuum model for plastic-brittle behavior of rock and concrete. Int. J Engineering Sciences, 17(1979): 121-37 [4] K. F. Ha. Basics of fractured physics. Science Press, Beijing, 2000 [5] B. B. Mandebrot. The fractal geometry of nature. W. H. Freeman and Co., San Francisco, Calif., 1982 [6] D. Krajcinovic and G. U. Fonseka. The continuous damage theory of brittle materials-Part I: general theory, ASME J Applied Mechanics, 48(1981): 809-15 [7] Zhang Lixiang, Zhao Zaodong, Li Qingbin. P-D-ε curve and cumulative damage of concrete in fatigue, J Engineering Mechanics, 19,5(2002): 81-97 [8] Li Qingbin, Zhang Chuhan and Wang Guanglun. Dynamic damage constitutive model of concrete in uniaxial tension, Engineering Fracture Mechanics, 53(1996): 449-55

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