A stochastic damage model for evaluating the internal deterioration of concrete due to freeze–thaw action

ORIGINAL ARTICLE

A stochastic damage model for evaluating the internaldeterioration of concrete due to freeze–thaw action

An Duan • Ye Tian • Jian-Guo Dai •

Wei-Liang Jin

Received: 9 August 2012 / Accepted: 29 May 2013

� RILEM 2013

Abstract This paper presents a stochastic damage

model for evaluating the internal deterioration of

concrete due to freeze–thaw action, which involves

great uncertainty and randomness. In this model, the

structural element of concrete is discretized into

infinite microelements, whose lifetimes are assumed

to be independent random variables. Then expressions

for the mean and variance of the damage of concrete

are analytically derived. To calibrate the model

parameters, a series of freeze–thaw tests in water on

non-air-entrained concrete were conducted and back-

calculation analyses were performed on the test results

of dynamic modulus. The reliability of the proposed

stochastic damage model is further validated through

comparisons with the results of 80 other existing test

specimens. The present model offers a theoretical

basis for exploring the statistical aspect of concrete

behavior during freeze–thaw.

Keywords Freeze–thaw action � Concrete �Micro-level � Stochastic damage model

1 Introduction

Frost damage is an important concern for concrete

infrastructures built in cold regions. There are two major

types of frost-induced concrete damage [7]: (1) the

internal micro-cracking and disruption of concrete

caused by frozen moisture, and (2) surface scaling,

which is local flaking or peeling of a finished surface as a

result of exposure to freeze and thawing [32]. The latter

is normally limited to the surface, whereas the former

leads to a substantial reduction in the mechanical

properties of concrete in terms of both its strength and

stiffness [24]. As a consequence, increasing attention

has been paid to the internal deterioration of concrete

due to frost damage over the past two decades. One of

the foci has been the modeling of the internal damage

evolution of concrete exposed to freeze–thaw action.

Attempts have been made to assess the frost-

induced internal damage of concrete using both

empirical [21, 22] and theoretical [3, 8] approaches.

In all of these models, however, the internal damage is

expressed in terms of the loss of dynamic modulus of

elasticity. Fagerlund [8] developed his model based on

the critical degree of saturation theory, whereas Cai

[3] deployed a hydraulic hypothesis and fatigue

damage theory to describe the damage process.

Nevertheless, most of these existing models are

deterministic, and as such they cannot reflect the

highly stochastic nature of concrete.

Concrete is, by nature, a heterogeneous material,

mainly because of its composite components. Such

A. Duan � Y. Tian (&) � W.-L. Jin

College of Civil Engineering and Architecture, Zhejiang

University, Hangzhou 310058, China

e-mail: [email protected]

J.-G. Dai

Department of Civil and Structural Engineering,

The Hong Kong Polytechnic University, Hung Hom,

Kowloon, Hong Kong

Materials and Structures

DOI 10.1617/s11527-013-0111-8

heterogeneity is also due to the physical and chemical

phenomena that occur during the manufacturing and

hardening of concrete and the micro-cracking that

results from drying and shrinkage. In addition, the

freeze–thaw damage of concrete is a very complicated

physical phenomenon that has not yet been fully

understood, despite the existence of various theories

explaining the mechanisms that lead to the damage.

When ice forms in the saturated pores of concrete,

tensile stresses are generated by the volumetric

increase (i.e., the water changes from a liquid to a

solid state) and the flow of water that is forced from the

pores [24]. These tensile stresses then cause the onset

and propagation of micro-cracks in the surrounding

concrete, resulting in the degradation of its mechanical

properties such as stiffness and strength. These

processes involve great uncertainty and randomness.

Therefore, it is more reasonable to adopt a stochastic

method rather than a deterministic one when studying

the frost damage evolution of concrete.

The stochastic method has long been adopted for

the durability assessment of concrete structures sub-

jected to carbonation or chloride ingress (e.g., [4, 30,

31]). However, few stochastic models have been

developed to predict the service life of concrete

structures exposed to frost environments [9, 29]. In

Fagerlund’s [9] model, the actual moisture content,

Sact, and the critical moisture content, SCR, in concrete

are treated as stochastic variables. The probability of

frost damage is then calculated by assuming a

triangular distribution for the probability density

function of Sact and SCR. Song and Ji [29] used the

Weibull distribution to analyze the reliability of frost

damage and to predict the residual life of concrete

after freeze–thaw exposure. Unfortunately, in these

two models, the damage evolution, which is a very

crucial indicator of the mechanical properties of

concrete under freeze–thaw action, is not discussed.

To evaluate the damage evolution of a brittle

material such as concrete, micromechanical modeling

is an effective approach. Because it not only provides

valuable insight into the damage evolution mecha-

nisms of concrete at the micro-level, but also facilitates

a quantitative correlation between them and the macro-

level behavior of concrete. In the micromechanical

modeling approach, quantities such as stress, stiffness

and damage represent the average behavior of collec-

tive micro-level elements in the given domain. The

most extensively used stochastic micromechanical

method is the parallel element method, which was

derived and extended to predict the damage propaga-

tion of a structural element under monotonic loading

[13, 14, 17]. In the parallel element method, the

structural element is idealized as a sequence of

individual springs (i.e., microelements) joined in

parallel. The failure strength (or failure strain) of each

microelement is assumed to be a random variable. It is

worth mentioning that Guan et al. [10] proposed a

micromechanics-based model to evaluate the evolu-

tion of the average internal freeze–thaw concrete

damage. However, in their model they did not consider

the stochastic characteristics of the damage, which

have been widely revealed by experimental

observations.

This study aims to develop a stochastic damage

model for concrete subjected to freeze–thaw action

that features both stochastic and micromodeling

characteristics. An experimental test program was

conducted to calibrate the parameters deployed in the

stochastic damage model. In addition, the validity of

the model is further demonstrated through comparison

with an extensive testing database.

2 Theoretical background of the stochastic

damage model for concrete subjected to frost

2.1 Model concept and hypotheses

Concrete suffers from gradient damage when sub-

jected to freeze–thaw action. Figure 1 represents the

two-dimensional plane of a structural concrete ele-

ment with a square cross section. The entire square

concrete domain can be discretized into infinite

microelements with random lifetimes. The small,

solid block inside the square represents a microele-

ment of concrete (Fig. 1) and is assumed to possess the

following characteristics:

Hypothesis 1 Each individual microelement has a

different lifetime, T. A microelement fails once the

duration of its exposure to freeze–thaw action, t,

exceeds its lifetime (i.e., t C T).

Hypothesis 2 The lifetimes of the microelements

are assumed to be independent random variables

following the three-parameter Weibull distribution,

which has the following cumulative distribution

function (CDF):


FT tð Þ ¼ 1� exp � gðt � cÞ½ �bn o

t� c

0 t\c

(ð1Þ

in which g is the scale parameter, ([0); b is the shape

parameter, ([0); and c is the location parameter of the

distribution, ([0) [1].

Hypothesis 3 The parameters used in the CDF of the

Weibull distribution to describe the lifetime of each

microelement are dependent on its position (x, y). For

the sake of simplicity, it is assumed that all of the

microelements with the same minimum distance to the

boundary of the domain have identical parameters.

The Weibull distribution has proven its applicabil-

ity in the lifetime prediction of concrete subjected to

freeze–thaw action [29]. In addition, the three param-

eters, g, b and c, provide the maximum flexibility in

defining different failure rates and thresholds for

different types of concrete exposed to various frost

environments. The shape parameter b can be regarded

as a constant because the shape of the probability

distribution function of the lifetime of concrete at all

locations should be similar if the concrete is treated as

a homogenous material. The parameter g indirectly

reflects the weakness of concrete because its increase

leads to an increase in the failure probability, whereas

the parameter c can be treated as the failure threshold

of concrete at a specific location because its increase

leads to a decrease in the failure probability. In the

following derivations, c is assumed to be inversely

proportional to g for mathematical simplicity:

c ¼ c1=g ð2Þ

in which c1 is a positive constant.

2.2 Damage evolution

2.2.1 Mean of the damage

The interior damage of the concrete element at an

exposure time t is represented by a random function

D(t) and can be defined as the ratio of the failed area to

the total area [16]:

DðtÞ ¼ AD

A0

¼ 1

A0

ZZ

XH t � Tðx; yÞð Þds ð3Þ

where T(x, y) is the random lifetime at the position (x,

y); AD is the failed area; A0 is the total area of the cross

section; X is the entire domain in Fig. 1 and given by

X = {(x, y) |-a B x B a, -a B y B a}; and H is the

Heaviside function, which can be written as

H xð Þ ¼ 0; x\0

1; x� 0

�ð4Þ

The mean damage computation is based on the

expected value of the damage function

lD tð Þ ¼ E D tð Þ½ �

¼ 1

A0

Z1

0

ZZ

XH t � sð ÞfT s ; x; yð Þdsds ð5Þ

in which fT(s; x, y) is the probability density function

of the lifetime T of the microelement at the position (x,

y), s is the integral variable, and ds = dxdy. Given the

definition of H, Eq. (5) can be transformed to

lD tð Þ ¼ 1

A0

ZZ

XFT t ; x; yð Þdxdy ð6Þ

in which FT(t; x, y) = Prob [T (x, y) \ t] is the CDF for

the lifetime T of the microelement at the position (x, y).

According to hypothesis 3, in the shaded triangular

area of Fig. 1, the FT(t; x, y) for the lifetime is only

related to x (i.e., independent of y) and is given by FT(t;

x), and the integral over domain X [in Eq. (6)] are

equal to the 8 times the integral over the shaded

triangular area. Thus, Eq. (6) can be rewritten as

lD tð Þ ¼ 8

A0

Z a

0

dx

Z x

0

FT t; xð Þdy ¼ 8

A0

Z a

0

xFT t; xð Þdx

ð7ÞBased on hypothesis 2, FT(t; x) has the form

FT t; xð Þ ¼ 1� exp � gðt � cÞHðt � cÞ½ �bn o

ð8Þ

Fig. 1 Idealized model of a concrete cross section


The concrete surface is more susceptible to freeze–

thaw cycles (FTCs) for the following reasons [24]: (1)

higher freezable water since the outer layer of concrete

can become saturated more quickly when the surface

is covered with water; (2) increased water/cement

ratio due to bleeding; (3) earlier shrinkage-induced

microcracking due to water evaporation; and (4) lager

time difference during the FTCs. On the other hand, it

is noticed that FT(t; x) increases with the increase of g,

meaning a more vulnerable microelement and a

shorter lifetime. Thus, the distribution of g along x

direction is assumed to follow the expression (Fig. 1):

g ¼ g0 þ c2x; 0� x� a ð9Þ

where g0 denotes the scale parameter at the center

point, and c2 represents the slope of the function g(x).

Both of these parameters are positive and serve as a

basis for the following derivation.

Substituting Eqs. (2) and (9) into (8) yields

FT t; xð Þ ¼ 1

� exp � g0 þ c2xð Þt � c1½ �H t � c1

g0 þ c2x

� �� b( )

ð10ÞCombining Eqs. (7) and (10), the expected value of

the damage can be obtained as follows:

lD tð Þ ¼ 1� 2

a2

Za

0

x exp � g0þ c2xð Þt� c1½ �H t� c1

g0þ c2x

� �� b( )

dx

ð11Þ

Let

G xð Þ ¼ x

exp � g0 þ c2xð Þt � c1½ �H t � c1

g0 þ c2x

� �� b( )

ð12Þ

Then

lD tð Þ ¼ 1� 2

a2

Za

0

GðxÞdx ð13Þ

2.2.2 Variance of the damage

The variance in the damage D (t) is

r2D tð Þ ¼ var D tð Þ½ � ¼ E D2 tð Þ

� �� l2

D tð Þ ð14Þ

where E[D2(t)] is the mean square of the damage and is

given by

E D2 tð Þ� �

¼Z1

0

Z1

0

1

A20

ZZ

X

ZZ

XH t � s1ð ÞH t � s2ð Þds1ds2

� �

� fT s1; s2; s1; s2ð Þds1ds2 ð15Þ

in which fT(s1, s2, s1, s2) is the joint probability density

function of the two random variables, T(s1) and T(s1),

at the positions of s1 (x1, y1) and s2 (x2, y2),

respectively. The expansion of Eq. (15) yields

E D2 tð Þ� �

¼ 1

A20

ZZ

X

ZZ

XFT t; t; s1; s2ð Þds1ds2 ð16Þ

where FT(s1, s2, s1, s2) = Prob [T(s1) B s1 \T(s2) B s2] = the joint distribution function of the

random variables T(s1) and T(s1).

Eqs. (14) and (16) yield the variance of the damage

parameter

r2D tð Þ ¼ 1

A20

ZZ

X

ZZ

XFT t; t; s1; s2ð Þds1ds2 � l2

D tð Þ

ð17ÞLee [15] developed a potentially useful bivariate

distribution for which the marginal distributions are

three-parameter Weibull distributions. Its survival

function is given by

F t1; t2ð Þ ¼ P T1 [ t1; T2 [ t2ð Þ

¼ exp � g1 t1 � c1ð ÞH t1 � c1ð Þ½ �b=rnn

þ g2 t2 � c2ð ÞH t2 � c2ð Þ½ �b=roro

ð18Þ

where the parameter r measures the conditional

association of T1and T2, and 0 \ r B 1.

Thus the joint distribution function FT(s1, s2, s1, s2)

can be rewritten as


F t1; t2ð Þ ¼P T1� t1; T2� t2ð Þ

¼1� exp � g1 t1 � c1ð ÞH t1 � c1ð Þ½ �bn o

� exp � g2 t2 � c2ð ÞH t2 � c2ð Þ½ �bn o

þ exp � g1 t1 � c1ð ÞH t1 � c1ð Þ½ �b=rnn

þ g2 t2 � c2ð ÞH t2 � c2ð Þ½ �b=roro

ð19ÞSubstituting Eqs. (5) and (19) into (17) leads to

r2D tð Þ¼ 1

A20

Z Z

X

Z Z

Xexpn�n

g1 t�c1ð ÞH t�c1ð Þ½ �b=r

þ g2 t�c2ð ÞH t�c2ð Þ½ �b=roro

ds1ds2 � 1�lD tð Þð Þ2

ð20Þ

2.3 Numerical approximation

Because the antiderivative of the function G(x) in Eq.

(13) cannot be explicitly obtained, a numerical method

is applied here to approximate the definite integralR a

0GðxÞdx. According to the trapezoidal rule (Burden

2004), the domain [0, a] is discretized into n equally

spaced panels and the approximation to this integral

becomes

Za

0

G xð Þ � a

n

G 0ð Þ þ G að Þ2

þXn�1

i¼1

G i� a

n

" #ð21Þ

When n tends toward infinity, the numerical results

approach illimitably to the true value of the integral.

Therefore, in the numerical analysis, the expected

value of the damage has the form

lD tð Þ ¼ 1� 2

a2

a

n

Gð0Þ þGðaÞ2

þXn�1

i¼1

G ia

n

" #ð22Þ

Similarly, the variance of the damage parameter

can be obtained as

r2D tð Þ ¼ 1

n4

Xn

i¼1

Xn

j¼1

Kij � 1� lD tð Þð Þ2 ð23Þ

in which Kij has the form

Kij ¼ exp � gi t � cið ÞH t � cið Þ½ �b=rnn

þ gj t � cj

� �H t � cj

� �� b=ro

ro ð24Þ

where gi = g0 ?c2(2i-1)a/2/n and ci = c1/gi.

Although the above derivation on the stochastic

damage of concrete is based on a square section, the

method is generic and would be suitable for sections

with other arbitrary shapes. In addition this approach

could be served as a basic framework for further

development, and may be expected to be applicable to

concretes with different constituents, but only if the

corresponding parameters g0, b, c1, c2 and r are

appropriately calibrated. Nevertheless, it should be

noted that the model has been derived based on some

hypotheses and simplifications. Thus, further study is

required and will be discussed in Sect. 4.

3 Model calibration and validation

3.1 Freeze–thaw tests

An experimental program was conducted to calibrate

the micromechanical parameters deployed in the above-

described stochastic damage model. In this program,

four groups of non-air-entrained concrete prismatic

specimens (100 mm 9 100 mm 9 400 mm) were pre-

pared with water to cement (w/c) ratios of 0.40, 0.45,

0.55 and 0.60. Each group consisted of three identical

specimens. The proportions of the concrete mixes are

presented in Table 1. In each mix, natural river sand and

crushed gravel with particle sizes varying from 5 mm to

16 mm were used as the fine and coarse aggregates,

respectively. The #425 (i.e., Chinese Standard) ordinary

Portland cement was used and its chemical composi-

tions are given in Table 2.

All of the specimens were demolded 48 h after the

casting and cured for 7 days in water, followed by

more curing in an air-conditioned laboratory. Before

the freeze–thaw tests, all of the specimens were again

immersed in water with a temperature of about 20 �C

for 4 days to maximize damage accumulation during

the freeze–thaw cycling tests. At the age of 28 days, the

concrete specimens submitted to rapid FTCs in water

were tested in accordance with the Chinese Standard

Test Method for the Durability of Normal Concrete

[11], which is almost the same as the Procedure A of

ASTM C666-92, with the exception that the freeze–

thaw testing begins when the concrete is 14 days old in

the latter case. Figure 2 illustrates the freeze–thaw test

setup. Each FTC consisted of a temperature decrease

from 8 to -17 �C, followed by a temperature increase

to 8 �C within about 3 h.


During the test, the average internal damage was

evaluated using the relative dynamic modulus of

elasticity (RDM) from the fundamental transverse

frequency data [11]. The total number of FTCs and the

frequency of RDM measurement depended on the

specimens’ deterioration rates. For the specimens in

Group F45 (i.e., with a w/c ratio of 0.45), the RDM

was measured once every 25 FTCs until 150 FTCs.

However, for the specimens in Group F60 (i.e., with a

w/c ratio of 0.6), the RDM was measured more

frequently (i.e., at 0, 10, 20, 30, 35, 45 and 50 cycles)

because the frost damage was much more severe due

Table 1 Mix proportions

and test resultsGroup code w/c Mix proportions (kg/m3) N D

Cement Fine

aggregate

Coarse

aggregate

Water D1 D2 D3

F40 0.40 466 653 1089 187 25 0.049 0.057 0.046

50 0.134 0.191 0.138

75 0.140 0.211 0.168

100 0.207 0.305 0.225

125 0.254 0.344 0.269

150 0.317 0.444 0.331

200 0.521 0.659 0.539

225 0.550 0.664 0.580

250 0.554 0.699 0.636

F45 0.45 433 587 1139 195 25 0.073 0.054 0.056

50 0.141 0.095 0.106

75 0.235 0.143 0.186

100 0.412 0.281 0.336

125 0.515 0.377 0.440

150 0.648 0.506 0.544

F55 0.55 360 647 1201 198 10 0.038 0.047 0.029

20 0.134 0.200 0.149

30 0.204 0.237 0.183

40 0.257 0.335 0.266

50 0.403 0.493 0.370

60 0.425 0.539 0.407

70 0.462 0.607 0.437

80 0.551 0.697 0.510

F60 0.60 305 720 1196 183 10 0.112 0.092 0.146

20 0.406 0.347 0.429

30 0.463 0.443 0.540

35 0.486 0.529 0.605

45 0.676 0.750 0.802

50 0.749 0.772 0.843

Table 2 Chemical compositions of the cement used (%, by weight)

Component SiO2 Al2O3 Fe2O3 CaO MgO SO3 Cl- Loss on ignition

Cement 28.28 5.42 3.11 55.95 1.63 2.06 0.014 2.5


to the high w/c ratio used in this group. The freeze–

thaw tests were stopped once the drop of RDM of

elasticity was larger than 0.6. The information for the

other two groups is included in Table 1.

3.2 Damage evaluation

The dynamic modulus of elasticity describes the

deformation resistance of concrete under impulsive

loads. The dynamic modulus of elasticity is

approximately equal to the initial tangent modulus

[20], and therefore is larger than the static modulus

of elasticity. According to Petersen [23], the former

is about 1.35 times the latter. The relative drop in

the dynamic modulus of elasticity can be treated as

the extent of the internal damage of concrete, as

follows:

D ¼ 1� Ed

E0

ð25Þ

where Ed and E0 denote the dynamic modulus of

elasticity of concrete after and before FTCs, respec-

tively. Ed/E0 is the RDM.

Table 1 presents a summary of the values of D for

each specimen after freeze–thaw exposure. Figure 3

presents the increase in the average damage lD with

the number of FTCs N, where lD is the average

damage of three identical specimens. These test results

were reliably obtained and are deemed to be suitable

for the following model development.

3.3 Calibration of the micromechanical

parameters

A multiple regression method [2] was deployed using

Matlab to calibrate the micromechanical parameters in

the stochastic model, as presented in Sect. 2. The

number of FTCs, N, was used instead of the variable

t in Eq. (12). Figure 4 presents the flowchart for

regressing the values of g0, b, c1, c2 and r. The initial

values of the five parameters were set at 0.002, 1.4,

0.02, 0.02 and 0.7, respectively. The analyses reveal

that the parameters b, c2 and r changed very little for

concrete with different w/c ratios, and it was easy to

achieve a convergence. Therefore, the b, c2 and r were

fixed at their converged values and the flowchart

shown in Fig. 4; that is, a two-parameter nonlinear

Fig. 2 Freeze–thaw test

setup

Fig. 3 Mean damage versus number of FTCs: experimental

results and model predictions


analysis procedure, was run once again. In general,

Eqs. (22) and (23) can be solved with sufficient

accuracy when n C 5. Table 3 presents a summary of

the micromechanical parameters calibrated from the

present tests.

As previously mentioned, b describes the form of

the density function of the Weibull distribution. For

concrete subjected to a similar deterioration environ-

ment, b hardly changes. b = 1.2 seems to be suitable

for normal concrete (with a w/c ratio of 0.4–0.6)

exposed to FTCs in water. The damage probability

was also found to increase with g0, which is the scale

parameter at the center of the specimen. A higher w/c

ratio of concrete led to a larger g0. The parameter c1,

which is the location parameter, represents a threshold

deterioration time beyond which the damage starts to

occur. This explains why concrete with a lower w/c

ratio corresponds to a larger value of c1. The parameter

c2 denotes the slope of the function g(x) and remained

constant within the tested range of the w/c ratio.

Finally, the parameter r had a great impact on the

variance of the damage. An increased r led to a

dramatic drop in the variance of the damage. A

constant value of 0.98 was found to be suitable for

r based on the analysis of the current test data.

Through the nonlinear regression analysis, the

relationship of two fluctuant parameters, g0 and c1,

for non-entrained concrete submitted to FTCs with the

w/c ratios can be expressed as follows:

g0 ¼ 1:7� 10�5 exp 22w/c� 6ð Þ þ 0:004 ð26Þc1 ¼ 0:98 exp �6:208w/cð Þ ð27Þ

Fig. 4 Flowchart for

calibrating the stochastic

damage model

Table 3 Model parameters calibrated from the present tests

w/c g0 b c1 c2 r

0.40 0.0043 1.2 0.081 0.01 0.98

0.45 0.0049 1.2 0.061 0.01 0.98

0.55 0.0116 1.2 0.030 0.01 0.98

0.60 0.0261 1.2 0.027 0.01 0.98


With Eqs. (26) and (27) and two constant values,

b = 1.2 and c2 = 0.01, the predicted evolution of the

mean frost damage of concrete with different w/c

ratios can be obtained as shown in Fig. 5. It is noted

that, for a given w/c ratio, there exists a threshold

number of FTCs, NCR, beyond which the concrete

starts to deteriorate (refer to the red line in Fig. 5).

This is because the initial moisture content of the

concrete specimen is below the critical moisture

content at the beginning of the FTC tests [8].

Consequently, no damage is induced during the initial

cycles. With further water uptake during the FTCs, the

threshold moisture content is reached in the whole or

parts of the specimen, resulting in the commencement

of frost damage. The proposed damage model effec-

tively captures such a mechanism. The value of NCR

depends on the water uptake during each cycle, in

addition to the initial moisture content in the concrete.

A higher NCR is needed for concrete with a lower w/c

ratio, which is characterized by a finer pore structure

and lower permeability.

The comparison between the predicted damage

evolutions and the experimental ones for the four types

Fig. 5 Predicted mean damage evolution for concrete with

different w/c ratios

Fig. 6 Evolution of the standard deviation of D: experimental data versus model prediction


of concrete mixtures tested are presented in Fig. 3,

which reveals that the model prediction agrees well

with the present experimental data (mean damage). Of

course, this is no surprise given that the model

parameters were calibrated from the test data. The

analytical curves for the standard deviation rD of the

damage with the number of FTSc are also presented in

Fig. 6, in comparison with their experimental coun-

terparts. Figure 7 further illustrates the mean lD,

lD ± rD and lD ± 2rD curves for the four different

w/c ratios studied. It shows that 77 % of the data lie

between lD - rD and lD ? rD, whereas 99 % of the

data lie between lD - 2rD and lD ? 2rD.

3.4 Further validation of the model

To further validate the proposed stochastic damage

model, a database of 80 other concrete specimens

subjected to FTCs (performed by other researchers)

was assembled. In building up the test database, the

following criteria were consistently applied to screen

the data:

(1) All of the specimens were subjected to rapid

freeze–thaw tests in water. The test procedures

were similar to those used in the testing

conducted in this study (Sect. 3.1)

(2) All of the specimens shared the following dimen-

sions: 100 mm 9 100 mm 9 400 mm, as per

the request of [11]

(3) The concrete was non-air-entrained and

incorporated no supplementary additives such

as fly ash or slag because this study focuses on

conventional concrete

The assembled database is summarized in Table 4,

where the predicted mean damage (obtained by using

the proposed stochastic model including these param-

eters calibrated by our own tests), lDmodel, and the

ratio of the predicted mean damage to its experimental

counterpart, lDmodel/lDtest, are also presented. The

Fig. 7 Analytically predicted variability of D–N relations


Table 4 Verification of the proposed stochastic damage model

Reference w/c Mix proportions (kg/m3) N lDtest lDmodel lDmodel/lDtest

Cement Fine

aggregate

Coarse

aggregate

Water

[27] 0.5 383 663 1154 193 25 0.15 0.081 0.54

50 0.18 0.209 1.16

75 0.28 0.332 1.19

100 0.38 0.443 1.17

[12] 0.5 N/A N/A N/A N/A 10 0.008 0.012 1.50

17 0.029 0.042 1.45

28 0.054 0.096 1.77

41 0.151 0.162 1.07

76 0.428 0.336 0.79

205 0.727 0.764 1.05

305 0.843 0.905 1.07

[3] 0.45 444 555 1200 200 25 0.075 0.039 0.52

50 0.130 0.133 1.02

75 0.200 0.229 1.15

100 0.325 0.321 0.99

125 0.475 0.407 0.86

0.60 333 636 1230 200 5 0.20 0.070 0.35

25 0.31 0.452 1.46

35 0.41 0.598 1.46

[21] 0.48 360 737 1106 172.8 50 0.10 0.170 1.70

100 0.27 0.382 1.41

150 0.78 0.556 0.71

[35] 0.5 360 769 1061 180 25 0.080 0.081 1.01

50 0.122 0.209 1.71

[34] 0.6 300 699 1191 180 10 0.15 0.171 1.14

25 0.50 0.452 0.90

[36] 0.4 360 700 1143 144 50 0.14 0.098 0.70

100 0.24 0.270 1.13

150 0.39 0.427 1.09

235 0.55 0.636 1.16

[19] 0.4 408 788 1043 163.2 25 0.074 0.017 0.23

50 0.095 0.098 1.03

75 0.124 0.185 1.49

100 0.187 0.270 1.44

125 0.331 0.352 1.06

150 0.488 0.427 0.88

175 0.639 0.496 0.78

200 0.745 0.559 0.75

[33] 0.4 525 525 1139 210 25 0.069 0.017 0.25

50 0.123 0.098 0.80

75 0.149 0.185 1.24


Table 4 continued


Cement Fine

aggregate

Coarse

aggregate

Water

100 0.179 0.270 1.51

125 0.246 0.352 1.43

150 0.303 0.427 1.41

175 0.345 0.496 1.44

190 0.398 0.534 1.34

200 0.458 0.559 1.22

0.45 456 660 1077 205 25 0.072 0.039 0.54

40 0.140 0.094 0.67

50 0.192 0.133 0.69

60 0.315 0.171 0.54

75 0.439 0.229 0.52

0.50 390 693 1131 195 25 0.086 0.081 0.94

50 0.200 0.209 1.05

75 0.253 0.332 1.31

100 0.329 0.443 1.35

115 0.404 0.503 1.25

125 0.467 0.540 1.16

0.55 345 714 1164 190 15 0.121 0.095 0.79

25 0.174 0.185 1.06

40 0.326 0.315 0.97

50 0.430 0.396 0.92

0.60 308 732 1194 185 10 0.072 0.171 2.38

15 0.224 0.271 1.21

25 0.363 0.452 1.25

30 0.479 0.530 1.11

40 0.691 0.659 0.95

[5] 0.48 406 630 1169 195 25 0.075 0.060 0.80

50 0.325 0.170 0.52

75 0.430 0.280 0.65

[6] 0.42 N/A N/A N/A N/A 25 0.12 0.025 0.21

50 0.17 0.108 0.64

75 0.26 0.198 0.76

100 0.36 0.284 0.79

125 0.47 0.366 0.78


Cement Fine aggregate Coarse aggregate Water

[28] 0.45 427 499 1284 192 25 0.077 0.039 0.51

50 0.216 0.133 0.62

75 0.380 0.229 0.60

0.55 360 611 1241 198 25 0.259 0.185 0.71

50 0.532 0.396 0.74


average value of lDmodel/lDtest is 1.01, reflecting a

good accuracy for the proposed model. The coefficient

of variation (i.e., 0.38) is relatively large, reflecting the

highly stochastic nature of the frost damage to the

concrete. Fig. 8 illustrates the comparison between the

model predictions and the test results. The agreement

seems to be acceptable at all levels of damage.

4 Discussion of the proposed model

The limitations of the proposed model should be noted

because the model parameters calibrated in this study

are only applicable for concrete that does not contain

air-entraining agents or other particular additives, such

as fly ash. In engineering practice, however, the use of

air-entraining agents and other pozzolanic materials

has become common practice. Additional tests and

analyses on a more extensive database are deemed

necessary, although the proposed stochastic damage

model is a generic one and can be extended for use

with other types of concrete.

Another issue to note is that the number of FTCs in

the specified accelerated laboratory exposure condi-

tion is used to represent the actual time scale when the

model parameters were calibrated in the proposed

model. It is obvious that a quantitative correlation

between the accelerated laboratory exposure and the

field exposure conditions is needed to facilitate

practical use of the proposed model. To find out how

many FTCs in the field represent each accelerated FTC

in the specified laboratory condition may prove a

challenging task and lies squarely outside the scope of

this study. Assuming that the model parameters for

different types of concrete are obtained and the issue

of environmental similarities between the laboratory

and field exposures is solved, the proposed stochastic

damage model, which is semi-empirical, could be used

in the service life prediction of concrete structures

subjected to freeze–thaw action.

The damage variable D is defined as the ratio of the

failed area to the total area [as shown in Eq. (3)].

Although this definition is widely accepted in the

continuum damage mechanics [16], it could not

capture all the physical and chemical processes

involved in the FTCs. Up to now, the frost deteriora-

tion mechanism is still debatable [8, 18, 25, 26] and

needs to be further studied for the future development

of this stochastic damage model.

In addition, the lifetimes of the microelements are

assumed to be independent random variables follow-

ing the three-parameter Weibull distribution [Eq. (1)],

which is an important basis for the whole deduction of

this stochastic model. The study of the true distribu-

tion of lifetime is required to develop a more general

and accurate model.

5 Conclusions

In this study, a stochastic damage model for concrete

subjected to freeze–thaw action has been developed. It

Fig. 8 lDmodel versus lDtes


offers a theoretical basis for evaluating the frost

damage to concrete with due consideration of the

features of frost damage along with its stochastic

nature. In the proposed model, the structural element

of concrete is assumed to be composed of infinite

microelements with random lifetimes, which follows

the three-parameter Weibull distribution. The expres-

sions for the mean and variance of the damage of

concrete are derived and an experiment program is

carried out to calibrate the model parameters. The

validity of the formed stochastic model is also

demonstrated by predicting 80 other existing tests.

Although the model parameters calibrated from this

study are only suitable for normal concrete with no air-

entraining agents or any other pozzolanic additives,

the proposed stochastic model is general and could be

extended to other types of concrete through a similar

procedure, which is a combined analytical-experi-

mental adaption. It is expected that the proposed

model can be deployed in the more reasonable

modeling of concrete frost damage with due consid-

eration of concrete’s stochastic nature.

Acknowledgments This research was financially supported

by the National Natural Science Foundation of China (No.

51108413), the Fundamental Research Funds for the Central

Universities (No. 2012QNA4016) and the Western Construction

Project of Ministry of Transportation (No. 20113288061110).

The authors are grateful to the staff and technicians in the lab for

their kind suggestions and assistance during the execution of this

research. In addition, the authors wish to thank The Hong Kong

Polytechnic University for this collaborative research

opportunity.

Appendix: derivations of the equations in Sect. 2.2

The derivation of Eq. (6) is as follows:

lD tð Þ ¼ 1

A0

Z t

0

ZZ

XH t � sð ÞfT s; x; yð Þdsds

¼ 1

A0

ZZ

XH t � sð Þ

Z t

0

fT s; x; yð Þdsds

¼ 1

A0

ZZ

X

Z t

0

fT s; x; yð Þdsds

¼ 1

A0

ZZ

XFT t; x; yð Þds

¼ 1

A0

ZZ

XFT t; x; yð Þdxdy

ð6Þ


lD tð Þ ¼ 8

A0

Z a

0

xFT t; xð Þdx¼ 2

a2

Z a

0

xFT t; xð Þdx

¼ 2

a2

Z a

0

x

�1� exp

��

g0þ c2xð Þt� c1½ �

�Hðt� c1

g0þ c2xÞ�b

dx

¼ 2

a2

Z a

0

xdx�Z a

0

x exp

(��

g0þ c2xð Þt� c1½ �

Hðt� c1

g0 þ c2xÞ�b)

dx

!

¼ 1� 2

a2

Z a

0

x exp

��

g0 þ c2xð Þt� c1½ �

Hðt� c1

g0 þ c2xÞ

�bgdx

ð11Þ


E D2 tð Þ� �

¼ 1

A20

ZZ

X

ZZ

X

Z t

0

Z t

0

fT s1; s2; s1; s2ð Þds1ds2

� �ds1ds2

¼ 1

A20

ZZ

X

ZZ

XFT t; t; s1; s2ð Þds1ds2

ð16Þ

References

1. Adnadevic B, Jankovic B, Kolar-Anic LJ et al (2007)

Normalized Weibull distribution function for modeling the

kinetics of non-isothermal dehydration of equilibrium

swollen poly (acrylic acid) hydrogel. Chem Eng J 130(1):

11–17

2. Allison PD (1999) Multiple regression: a primer. Pine Forge

Press, Thousand Oaks

3. Cai H (1998) Prediction model of concrete freeze-thaw

durability. Ph.D. dissertation, Tsinghua University, Bejing,

China (in Chinese)

4. Ciampoli M (1999) A probabilistic methodology to assess

the reliability of deteriorating structural elements. Comput

Methods Appl Mech Eng 168(1–4):207–220

5. Cheng HQ, Gao DY (2011) Experimental study on damage

of polypropylene fiber concrete in freeze-thaw cycles.

J Southeast Univ (Natural Science Edition) 40(su-

pII):197–200 in Chinese

6. Cheng HQ, Zhang LS, Li PX (2003) The influence of

freeze–thaw to concrete strength. Henan Sci 21(2):214–216

in Chinese

7. Fagerlund G (1997) Internal frost attack: State of the art.

Frost Resistance of Concrete. In: 2nd International RILEM

Workshop on Frost Resistance of Concrete, Essen, Ger-

many, pp 157–172


8. Fagerlund G (2004) A service life model for internal frost

damage in concrete. Lund Institute of Technology, Lund

9. Fagerlund G (1999) Service life with regard to frost attack- a

probabilistic approach. In: Lacasse MA, Vanier DJ (eds)

Proceedings of the Eighth International conference on

Durability of Building Materials and Components. Van-

couver, May 30–June 3, pp 1268–1277

10. Guan YG, Sun W, Miu CW (2001) One service life pre-

diction model for the concrete based on the reliability and

damage theories I: narration and establishment of the model.

J Chin Ceram Soc 29(6):530–534 in Chinese

11. GBJ82-85, Chinese Standard, Test method for durability of

normal concrete (1997) China Architecture and Building

Press, Beijing (in Chinese)

12. Hasan M, Nagai K, Sato Y et al (2002) Tensile stress-crack

width model for plain concrete damaged by freezing and

thawing action. Proc Jpn Concr Inst 24(2):109–114

13. Kandarpa S, Kirkner DJ, Spencer BF (1996) Stochastic

damage model for brittle materials subjected to monotonic

loading. J Eng Mech 126(8):788–795

14. Krajcinovic D, Silva MAG (1982) Statistical aspects of the

continuous damage theory. Int J Solids Struct 18(7):557–

562

15. Lee L (1979) Multivariate distribution having Weilbull

properties. J Multivar Anal 9(2):267–277

16. Lemaitre J, Desmorat R (2005) Engineering damage

mechanics. Springer, New York

17. Li J, Ren XD (2011) Stochastic damage model for concrete

based on energy equivalent strain. Int J Solids Struct

46(11–12):2407–2419

18. Litvan GG (1980) Freeze-thaw durability of porous building

materials. In: Sereda PJ, Litvan GG (eds) Durability of

building materials and components. Special Publication

STP 691, American Society for Testing and Materials,

Philadelphia, pp 455–463

19. Liu YM (2010) Freeze-thaw resistance of concrete of the

research and preparation. Master’s Dissertation, Chang’an

University, Xi’an (in Chinese)

20. Mindess S, Young J (1981) Concrete. Prentice-Hall,

Englewood Cliffs, p 671

21. Ning ZJ, Ba HJ, Yang YZ (2009) Experimental study of

fracture damage in concrete subjected to freeze-thaw cycles.

J Harbin Eng Univ 30(1):27–32 (in Chinese)

22. Penttala V (2006) Surface and internal deterioration of

concrete due to saline and non-saline freeze–thaw loads.

Cem Concr Res 36(5):921–928

23. Petersen L, Lohaus L, Polak MA (2007) Influence of

freezing-and-thawing damage on behavior of reinforced

concrete elements. ACI Mater J 104(4):369–378

24. Pigeon M, Pleau R (1995) Durability of concrete in cold

climates. E and FN Spon, London, pp 23–125

25. Powers TC (1975) Freezing effect in concrete. American

Concrete Institute, SP-47, Detroit, pp 1–12

26. Scherer GW (1999) Crystallization in pores. Cem Concr Res

29(8):1347–1358

27. Shang HS, Song YP (2006) Experimental study of strength

and deformation of plain concrete under biaxial compres-

sion after freezing and thawing cycles. Cem Concr Res

36(10):1857–1864

28. Shang HS, Song YP (2007) Experimental investigation on

different w/c concrete under biaxial compression after

freeze-thaw cycles. J Dalian Univ Technol 47(6):862–866

(in Chinese)

29. Song YP, Ji XD (2006) Analysis on reliability of concrete

under freezing thawing action and evaluation of residual

life. J Hydraul Eng 37(3):259–263 (in Chinese)

30. Sudret B, Defaux G, Pendola M (2007) Stochastic evaluation

of the damage length in RC beams submitted to corrosion of

reinforcing steel. Civil Eng Environ Syst 24(2):165–178

31. Teply B, Chroma M, Rovnanık P (2010) Durability assess-

ment of concrete structures: reinforcement depassivation

due to carbonation. Struct Infrastruct Eng 6(3):317–327

32. Valenza JJ, Scherer GW (2007) A review of salt scaling: I.

Phenomenology. Cem Concr Res 37(7):1007–1021

33. Zhang HY (2009) The research of frost-resisting durability

of concrete. Master’s Dissertation, Inner Mongolia Uni-

versity of Science and Technology, Baotou (in Chinese)

34. Zhang P, Zhang LS, Zhao TJ et al (2011) Water absorption

properties of concrete after freeze–thaw damages. J Build

Mater 14(2):155–159 (in Chinese)

35. Zhang Y (2007) Study on mesostructure and freeze-thaw

resistance about concrete material. Master’s Dissertation,

Lanzhou Jiaotong University, Lanzhou (in Chinese)

36. Zhao XL, Wei J, Huang YY (2002) Relationship between pore

structure change of concrete and its frost durability degradation.

J Wuhan Univ Technol 24(12):14–17 (in Chinese)


Documents

A stochastic damage model for evaluating the internal deterioration of concrete due to freeze–thaw action