The distribution state evolution of concrete cracking evolution energy has been discussed in which with dissipative system characteristics is considered, and combined the theory of information entropy with energy method. The function of entropy evolution change for in different stage of crack stable and unstable propagations evolution is established. The element damage extent formula is deduced, which can be applied to judge the stage of crack. Finally, the cracking process of double span continuous beam is simulated by Midas/FEA to compare with other literature. The result shows that the strain energy entropy function proposed can is be capable of well describing the evolution law of concrete cracking evolution

Bin Zhao

Changsheng Xiang

Fuchao Mao

Liang Zhang

Yu Zhou

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 217 
Research of concrete cracking propagation based on 
information entropy evolution 
Changsheng Xiang1, Yu Zhou2, Bin Zhao3, Liang Zhang4, Fuchao Mao5 
1, 2, 5Key Laboratory of Disaster Prevention and Mitigation in Civil Engineering of Gansu Province, 
Lanzhou University of Technology, Lanzhou, China 
3Qingshui County Transportation Bureau, Tianshui, Gansu, P. R. China 
4School of Civil Engineering, Lanzhou University of Technology, Lanzhou, Gansu, P. R. China 
1Corresponding author 
E-mail: 1xiangcs@lut.cn, 2yuzhou923@outlook.com, 3531506309@qq.com, 4lzssnhuast@163.com, 
5maofuchao@foxmail.com 
(Received 9 November 2016; accepted 14 November 2016) 
Abstract. The distribution state evolution of concrete cracking evolution energy has been 
discussed in which with dissipative system characteristics is considered, and combined the theory 
of information entropy with energy method. The function of entropy evolution change for in 
different stage of crack stable and unstable propagations evolution is established. The element 
damage extent formula is deduced, which can be applied to judge the stage of crack. Finally, the 
cracking process of double span continuous beam is simulated by Midas/FEA to compare with 
other literature. The result shows that the strain energy entropy function proposed can is be capable 
of well describing the evolution law of concrete cracking evolution. 
Keywords: concrete cracking, dissipative structural, entropy evolution, strain energy method, 
information entropy. 
1. Introduction 
Entropy was proposed by German famous physicist Clausius in 1865, which is a special 
parameter in thermodynamics and statistical physics. As a state variable, the change of entropy 
was related to the dispersion of matter [1], and the macroscopic significance indicates the uniform 
degree of the energy distribution of the system. In an isolated system, the occurrence of any 
irreversible process will lead to the rising of entropy [2]. At present, for non-reversible process, it 
always goes along a single direction of entropy increasing and eventually reaches equilibrium 
while entropy reaches maximum [3-5]. 
In recent years, the research of concrete crack expansion was caused great concern in civil 
engineering. The discrete crack model and distribution crack model are coupled by Wells G. N. 
to simulate the crack evolution which includes micro-cracking, crack propagation and final 
destruction for a unilateral cut concrete specimens [6, 7]. The fracture theory and analysis method 
of concrete crack development are established by Shi-lang Xu and Yan-hua Zhao, the double-K 
fracture criterion was proposed according to experimental result [12]. The concrete crack 
propagation with the perspective of energy distribution was analyzed by Pei-jiang Cong and 
Chong-shi Gu [2]. However, double-K fracture criterion proceeding from the linear elastic fracture 
mechanics and the study stage of strain energy entropy still remains on identification of crack 
initiation and stable growth. In this paper, crack evolution function based on entropy was proposed 
in the different stage of craching propagation, and the whole entropy change processing of 
concrete crack is studied based on information entropy and energy method to prove the validity 
of the formula. 
2. Theory of information entropy  
The strain energy of unit ݆ by the finite element theory [9]: 
ݍ௝ =
1
2 ෍ {ߪ
௘}்{ߝ௘}ݒ݋݈௞ + ܧ௘௣
௡௜௡௧
௞ୀଵ
, (1)
RESEARCH OF CONCRETE CRACKING PROPAGATION BASED ON INFORMATION ENTROPY EVOLUTION.  
CHANGSHENG XIANG, YU ZHOU, BIN ZHAO, LIANG ZHANG, FUCHAO MAO 
218 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
where, ݍ௝ is the strain energy of unit ݆; ݊݅݊ݐ – the number of unit integration points; ߪ௘ – the stress 
vector; ߝ௘ – the elastic strain vector; ݒ݋݈௞ – the volume of unit integration point ݇; ܧ௘௣ – plastic 
strain energy of units. Then the total strain energy ܳ is shown as following: 
Q= ෍ qj
N
j=1
, (2)
where, ܰ is the number of units. Set: 
ߣ௝ =
ݍ௝
ܳ ,     ෍ ߣ௝
ே
௝ୀଵ
= 1, ߣ௝ ≥ 0, ሺ݆ = 1,2, . . . , ݊ሻ. (3)
The information entropy function of the structure is defined as: 
ܵ = −߮ ෍ ߣ௝lnߣ௝
ே
௝ୀଵ
, (4)
where, ߮ is a constant coefficient, and set as 1.0 in engineering. 
3. Crack evolution function based on entropy 
3.1. Crack initiation and stable growth stage 
Based on the finite element theory and the characteristic of dissipative system [2], the change 
of strain entropy can be described by Eq. (6) as following: 
Δܵ = ܵ଴ − ௜ܵ,     ௜ܵ = −߮ ෍ ቆ
ݍ௝௜
∑ ݍ௝௜ே௝ୀଵ
ቇ ln
ே
௝ୀଵ
ቆ ݍ௝௜∑ ݍ௝௜ே௝ୀଵ
ቇ, (5)
where, ܵ଴ – the original entropy of structure; ௜ܵ – strain energy entropy of structure for the ݅th step; 
ݍ௝௜ – strain energy of unit ݆ for the ݅th step. The structural entropy difference of strain energy ܵ at 
different load step is obtained by Eq. (5). When ܵ = 0, there is no damage in structure; When  
ܵ ≠ 0, damage occurs. 
3.2. Failure stage 
3.2.1. Energy consumption of fracture zone 
When the external load ܲ reaches its maximum ௠ܲ௔௫, crack will expand rapidly. According to 
softening distribution cohesion model on the fracture zone [11], the work that cohesion act on unit 
area at the critical state of instability is [13]: 
ܩூ௖௖ =
1
ܽ௖ − ܽ଴ ׬ Γሺݔሻ݀ݔ௔೎௔బ
, (6)
߁ሺݔሻ = න ߪሺݓሻ
௪ೣ
଴
݀ݓ, (7)
where, ݔ – the horizontal distance between any point and the crack gap at fracture zone; ܽ଴ is the 
initial length of the gap; ܽ௖ – the critically instable length; Γሺxሻ  the fracture energy at ݔ, that is 
the absorbed energy when the gap of crack increases from 0 to ݓ௫.  
RESEARCH OF CONCRETE CRACKING PROPAGATION BASED ON INFORMATION ENTROPY EVOLUTION.  
CHANGSHENG XIANG, YU ZHOU, BIN ZHAO, LIANG ZHANG, FUCHAO MAO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 219 
3.2.2. Entropy function establishment of failure stage 
According to the principle of conservation of energy, the energy balance equation is: 
ܧ௧௢௧ = ߁ሺݔሻ + ܳ௜௡௘, (8)
where, ܧ௧௢௧ – the total energy of structure; ܳ௜௡௘ – strain energy of structure. 
௝ܵ଴ is described as the entropy of unit ݆ before structure being disturbed. According to the 
Eq. (6), the fracture criterion of crack is established: 
ܥ = ௝ܵబ − ܵௗ௝ ቄ= 0,≠ 0, (9)
where, ܵௗ௝ – the structural strain energy entropy for disturbed structure. When ܥ = 0, the unit ݆ of 
structure is not disturbed (no cracking). When ܥ ≠ 0, the damage extent is defined as: 
௝ܿ = ቤ ௝ܵబ
− ܵௗ௝
௝ܵబ
ቤ × 100 %. (10)
Judge further whether the unit is invalid according to the literature [8, 14, 15]. Using the 
literature [8] and Eq. (9) to determine the number of destructive units Num. Then structural strain 
energy ݍௗ௝ of unit ݆ for the ݅th step can be obtained after the crack appearing, the Eq. (7) can be 
further expressed: 
௜ܵ = − ෍
ݍௗ௜௝
ܳ௜௡௘௜ ln
ݍௗ௜௝
ܳ௜௡௘௜
ேିே௨௠
௝ୀଵ
, (11)
where, ݍௗ௜௝ – the strain energy of unit ݆ for the ݅th step after crack appearing; ܳ௜௡௘௜ – strain energy 
of structure for the ݅th step. 
3.3. Numerical calculation 
A concrete beam model is established by Midas/FEA to analyze crack evolution. Stress-strain 
data is extracted to calculate strain energy of unit which can be used to judge whether unit 
damaged or not. If damaged, the unit will be eliminated in coming analysis. Meanwhile, the 
fracture energy is calculated. Then the strain energy of structure should be recalculated for the 
shorten FE model, and the strain energy entropy will be obtained by Eq. (11), correspondingly. 
Fig. 1 shows the specific process. 
4. Numerical computation 
4.1. Numerical model 
A two-equal-span continuous concrete beam model is established by Midas/FEA. Each span 
is 1000 mm, and cross-section size is 400 mm×200 mm, and the strength of concrete is C50. The 
surface load area at the first mid-span is 400 mm×10 mm and the load intensity is 20 N/mm2 
(Fig. 2). The system automatically load steps by Midas/FEA is applied for analysis and control, 
and the initial load factor is 0.01. Material properties are shown in Table 1. 
There are two major cracks at first mid-span and mid-support (Fig. 3) by running the nonlinear 
analysis module of cracking of Midas/FEA. And while load increasing, the crack evolution at the 
first mid-span will contain crack initiation, stable growth, unstable propagation and failure stage. 
RESEARCH OF CONCRETE CRACKING PROPAGATION BASED ON INFORMATION ENTROPY EVOLUTION.  
CHANGSHENG XIANG, YU ZHOU, BIN ZHAO, LIANG ZHANG, FUCHAO MAO 
220 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
 
Fig. 1. The chart of entropy function numerical computation 
 
Fig. 2. Midas/FEA beam model 
Table 1. Material properties 
Elastic 
modulus 
(N/mm2)
Poisson’s 
ratio 
Weight 
density 
(N/mm3)
Influence 
of 
transverse 
crack 
Constraint 
effect 
34965 0.167 2.5×10-5 Vecchio Collins 
Selby 
vecchio 
 
 
 
a)  b) 
Fig. 3. Crack state: a) at first middle span, b) at middle support 
4.2. Calculation of crack evolution entropy 
4.2.1. Entropy changes of first crack initiation and stable growth 
The structure strain energy entropy of every step is calculated, and entropy difference of strain 
energy can be obtained by initial value and current one (Table 2). Table 2 shows when incremental 
step is between 1 and 7, ܵ = 0, which means the structure is undamaged, while at 8th increment 
step, first crack appears and entropy difference of strain energy continues to increase. 
The unit entropy difference at the first mid-span is calculated, the curve of step and entropy 
difference is drawn in Fig. 4. From the curve, at 8th step, the entropy difference is changing, which 
means crack appears, this conclusion is consistent with integral analysis result in Table 2. While 
till 19th step, entropy difference of strain energy is stable growth, which means that this crack is 
under stable extending. The 19th step is instable point of cracking.  
RESEARCH OF CONCRETE CRACKING PROPAGATION BASED ON INFORMATION ENTROPY EVOLUTION.  
CHANGSHENG XIANG, YU ZHOU, BIN ZHAO, LIANG ZHANG, FUCHAO MAO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 221 
Table 2. Change for load-entropy incremental step 
No. Load factor 
Strain energy 
entropy (J) 
Entropy 
difference (Δ ௜ܵ) No.
Load 
factor 
Strain energy 
entropy (J) 
Entropy 
difference (Δ ௜ܵ) 
1 0.010000 5.989 0 11 0.314375 6.051 0.062 
2 0.020000 5.989 0 12 0.352344 6.075 0.086 
3 0.035000 5.989 0 13 0.390312 6.094 0.105 
4 0.050000 5.989 0 14 0.428281 6.117 0.128 
5 0.072500 5.989 0 15 0.466250 6.138 0.149 
6 0.095000 5.989 0 16 0.504219 6.159 0.170 
7 0.128750 5.989 0 17 0.542188 6.179 0.190 
8 0.162500 5.991 0.002 18 0.580156 6.201 0.212 
9 0.213125 6.002 0.013 19 0.618125 6.222 0.233 
10 0.263750 6.024 0.035 20 0.656094 6.239 0.250 
4.2.2. Entropy changing of second crack initiation 
Mid-support stress-strain data is extracted and the element strain energy is solved to calculate 
entropy difference, the result is drawn in Fig. 5. The mid-support appears new crack and entropy 
difference of strain energy is nonzero when loading to the 12th increment step (Fig. 4). 
 
Fig. 4. Strain energy entropy difference  
at the first mid-span 
 
Fig. 5. Strain energy entropy difference  
for mid-span 
4.2.3. Entropy changing of failure stage 
When up to the 19th increment step, the first crack shows unstable state. After determining the 
number of invalid unit, the strain energy of structure is recalculated to obtain the entropy for failure 
stage. At the same time, entropy changing with calculation method proposed by the literature [2] 
(Table 3) is compared with Table 2, the comparison curve is drawn in Fig. 6. 
The data of this paper indicates that strain energy entropy decreases at first and then increase 
when incremental step from 20 up to 25. It is due to the instability of crack at first mid-span and 
energy redistribution happening. Part of energy is consumed by the development of new crack. 
Strain energy entropy function is a convex function defined on a convex set, however, 
according to the data of the literature [2], strain energy entropy will be continually increasing after 
first crack appearing. Therefore, the data here is more accurate. 
According to above analysis, the rule of crack evolution is as following: the strain energy 
entropy remains stable before crack appearing, and starts to increase after crack appearing, then 
reaches instable point of cracking, and decreases at first then rises until concrete failure. 
5. Conclusions 
Firstly, a brief review about basic theory of entropy is introduced, and then the strain energy 
entropy is proposed based on dissipative system and energy method, which being considered with 
RESEARCH OF CONCRETE CRACKING PROPAGATION BASED ON INFORMATION ENTROPY EVOLUTION.  
CHANGSHENG XIANG, YU ZHOU, BIN ZHAO, LIANG ZHANG, FUCHAO MAO 
222 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
the internal energy distribution. The changing of entropy is studied under the whole concrete crack 
evolution stage. The results show that: 
1) Strain energy entropy function is a convex function. Therefore, the data in this paper is more 
accurate by being compared with that of the literature [2]. 
2) The crack initiation, stable growth, unstable propagation and failure stage can be described 
by the whole curve of strain energy entropy changing, which is proved more comprehensive by 
the numerical simulation of continuous beam. 
3) The new thought based on whole curve of strain energy entropy changing, can offer a new 
method to research concrete cracking evolution. 
Table 3. Entropy changing for instable stage 
incremental step 
 Load factor 
Strain energy 
entropy (J) 
Data of the literature 
(Cong et al. 2009) 
21 0.694062 6.230 6.253 
22 0.732031 6.226 6.266 
23 0.770000 6.238 6.279 
24 0.807969 6.260 6.292 
25 0.845937 6.280 6.303 
26 0.883906 6.291 6.321 
27 0.921875 6.294 6.338 
28 0.959844 6.297 6.355 
 
 
Fig. 6. Strain energy entropy  
for crack evolution’s comparison figure 
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Research of concrete cracking propagation based on information entropy evolution

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Research of concrete cracking propagation based on information entropy evolution

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