A fractional order rate approach for modelingconcrete structures subjected to creep and fracture
F. Barpi
*
, S. Valente
1
Department of Structural and Geotechnical Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Received 24 March 2003; received in revised form 12 December 2003
Abstract
The paper analyses the behaviour of concrete in the case of quasistatic fracture. The attention is focused on theinteraction between strainsoftening and timedependent behaviour: a viscous rheological element (based on a fractional order rate law) is coupled with a micromechanical model for the fracture process zone. This approach makes itpossible to include a whole range of dissipative mechanisms in a single rheological element. Creep fracture in mode Iconditions is analysed through the ﬁnite element method and the cohesive (or ﬁctitious) crack model. The comparisonwith creep tests executed on threepoint bending conditions (three diﬀerent load levels) shows a good agreement both interms of failurelifetime, and, load–displacement.
2004 Elsevier Ltd. All rights reserved.
Keywords:
Cohesive; Concrete; Crack; Creep; Fracture; Fractional; Longterm behaviour; Softening; Viscosity
1. Introduction
The longterm performance of concrete structures is fundamentally aﬀected by the behaviour of thematerial after cracking. It is well known that concrete presents a diﬀused damage zone within which microcracking increases and stresses decrease as the overall deformation increases. This results in the softening of the material in the so called
fracture process zone
(FPZ), whose size can be compared with acharacteristic dimension of the structure. This dimension is not constant and can vary during the evolutionary process. In this context, a numerical method (based on ﬁnite or boundary elements) has to be usedtogether with the
cohesive
or
ﬁctitious
crack model as shown by Barenblatt (1959), Dugdale (1960) andHillerborg et al. (1976).The interaction between strainsoftening and timedependent behaviour is analysed, with the emphasison very slow or quasistatic fracture. This is the case of cracking in massive concrete structures like dams,where inertial forces can be neglected. In this ﬁeld three approaches will be considered. The ﬁrst is based on
*
Corresponding author. Tel.: +39115644886; fax: +39115644899.
Email addresses:
fabrizio.barpi@polito.it (F. Barpi), silvio.valente@polito.it (S. Valente).
1
Tel.: +39115644853; fax: +39115644899.00207683/$  see front matter
2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijsolstr.2003.12.025International Journal of Solids and Structures 41 (2004) 2607–2621www.elsevier.com/locate/ijsolstr
Nomenclature
e
,
e
1
deformations of the rheological model
r
stress of the rheological model, stress in the cohesive zone
r
1
stress of the rheological model
E
1
,
E
2
Young
s moduli of the rheological model
g
classical Newton
s viscosity parameter
s
1
¼
g
E
1
relaxation time
D
a
ðÞ ¼
d
a
ðÞ
d
t
a
fractional derivative operator of order
aa
order of diﬀerentiation
C
Euler
s Gamma function
C
ð
x
Þ ¼
R
1
0
e
i
t
ð
x
1
Þ
d
t
¼
lim
n
!1
n
x
n
!
x
ð
x
þ
1
Þð
x
þ
n
Þ
y
ð
t
Þ
generic function of time
U
1
a
ð
t
Þ
kernel of the noninteger diﬀerentiation deﬁnition
b
i
ð
a
Þ
i
th coeﬃcient of the numerical approximation of the noninteger derivative
E
c
concrete Young
s modulus
K
1
,
K
2
elastic constants of the rheological model (see Fig. 6)
m
Poisson
s ratio
G
F
fracture energy (area below the curve of Fig. 5)
f
t
ultimate tensile strength
r
F
:
C
:
T
maximum principal (tensile) stress acting at the ﬁctitious crack tip
w
crack opening displacement (also called COD)COD crack opening displacement (also called
w
)
w
c
critical crack opening displacement (beyond
w
c
no stresses are transferred in the cohesive zone)
V
f
aggregate volume fraction
K
homIc
fracture toughness of the homogenized material
b
concrete microstructural parameter
b
¼
ð
K
homIc
Þ
2
E
c
ð
1
V
f
Þ
f
t
D
r
stress relaxation due to creep
D
w
creep displacement
t
time
z
distance measured from the bottom of the specimen
D
t
time stepd
r
t
stress relaxation computed in each point of the FPZ (depends on local conditions only becauseit is assumed
w
¼
const)d
w
t
creep displacement computed in each point of the FPZ (depends on local conditions onlybecause it is assumed
r
¼
const)d
r
real stress increment in the FPZ (depends on global and local conditions)d
w
real displacement increment (depends on global and local conditions)
H
specimen height
P
max
maximum (or peak) load
P
cost
constant load level during the creep phase
K
T
positive deﬁnite tangential stiﬀness matrix
C
T
negative deﬁnite tangential stiﬀness matrix
P
external load vector
D
k
load multiplier
Q
unbalanced load vector
D
u
displacement vector
2608
F. Barpi, S. Valente / International Journal of Solids and Structures 41 (2004) 2607–2621
the concept of activation energy and ratedependent softening that has been developed in a series of paperby Bazant and coworkers (Ba
zant, 1992; Ba
zant and Gettu, 1992; Ba
zant and Jir
asek, 1992, 1993; Wu andBa
zant, 1993). This method was recently modiﬁed by van Zijl et al. (2001). The second approach is based onthe inclusion of a standard rheological model for creep and relaxation into the ﬁctitious crack model inorder to accommodate the time dependency of crack opening, the latter in some instances being establishedby ﬁtting stress relaxation results (Hansen, 1990, 1991; Zhou and Hillerborg, 1992; Zhang and Karihaloo,1992a,b; Carpinteri et al., 1995, 1997; Barpi et al., 1999a). The third approach combines a micromechanicalmodel for the static softening behaviour of cracked concrete in the fracture process zone (Huang and Li,1989) with a rheological model for the timedependent concrete behaviour (Santhikumar and Karihaloo,1996, 1998; Santhikumar et al., 1998).In the present paper the third approach is enhanced using a
fractional order rate
law and is applied to thenumerical simulation of the threepoint bending tests described by Zhou (1992).
2. Description of the rheological model
Rheology is concerned with timedependent deformation of solids. In the simplest rheological model of the linear standard viscoelastic solid (Fig. 1), the springs are characterized by linear stress–displacementrelationships:
r
1
¼
E
1
ð
e
e
1
Þ
;
ð
1a
Þ
r
2
¼
E
2
e
:
ð
1b
Þ
In this paper, the dashpot is based on the following
fractional order rate
law for the internal variable
e
1
:
D
a
e
1
¼
d
a
e
1
d
t
a
¼
r
1
E
1
s
a
1
¼
e
e
1
s
a
1
with
a
2 ð
0
;
1
Þ
;
ð
2
Þ
where the fractional diﬀerentiation of a function
y
ð
t
Þ
is deﬁned according to Oldham and Spanier (1974)and Carpinteri and Mainardi (1997). Eq. (2) represents a generalization of the wellknown Newton
sconstitutive law for the dashpot
ð
r
¼
g
d
e
d
t
Þ
.In particular
D
ð
1
a
Þ
y
ð
t
Þ ¼
Z
t
0
U
1
a
ð
t
t
Þ
y
ð
t
Þ
d
t
;
ð
3
Þ
E
σεε

εσ
2
E
1
1 1
Fig. 1. Rheological model.
F. Barpi, S. Valente / International Journal of Solids and Structures 41 (2004) 2607–2621
2609
where
U
1
a
ð
t
Þ ¼
t
a
þ
C
ð
1
a
Þ
with
t
þ
¼
t
if
t
>
00 if
t
<
0
:
ð
4
Þ
In the previous expression
C
represents the
Gamma function
. Eq. (3) can also be obtained by using anhereditary model based on a Rabotnov fractional exponential kernel (see Karihaloo, 1995).A convergent expression for the
a
order fractional derivative operator
D
a
is given by
D
a
y
ð
t
Þ ¼
D
1
D
ð
1
a
Þ
y
ð
t
Þ ¼
dd
t
Z
t
0
U
1
a
ð
t
t
Þ
y
ð
t
Þ
d
t
¼
1
C
ð
1
a
Þ
dd
t
Z
t
0
y
ð
t
Þð
t
t
Þ
a
d
t
:
ð
5
Þ
In the case of
a
¼
1 the classical dashpot with an integer order rate law is obtained from Eq. (2). Inparticular, the solutions for the relaxation problem (under constant
w
) and for the creep problem (underconstant
r
) become of exponential type, with
s
1
as the
relaxation time
, and
s
1
E
1
þ
E
2
E
2
as the
retardation time
.Response diagrams are plotted in Figs. 2 and 3 (see Barpi and Valente, 2003).
10
3
10
2
10
1
10
0
10
1
10
2
10
3
Nondimensional time
t
/
τ
1
0.50.60.70.80.91
N o n d i m . s t r e s s r e l a x a t i o n
σ / σ
(
t
= 0 )
α=0.5α=0.3α=1.0
Fig. 2. Stress relaxation functions.
10
3
10
1
10
1
10
3
Nondimensional time
t
/
τ
1
11.21.41.61.82
N o n d i m e n s i o n a l d i s p l a c e m e n t
w / w
(
t
= 0 )
α=0.5α=0.3α=1.0
Fig. 3. Creep displacement functions.2610
F. Barpi, S. Valente / International Journal of Solids and Structures 41 (2004) 2607–2621
Fig. 4 shows the the inﬂuence of the noninteger derivative on the creep rate (i.e., the derivative withrespect to time of the creep functions). This ﬁgure represents another way to show the diﬀerence betweenthe model based on integer derivative (straight line) and the model based on a noninteger derivative(dashed and dasheddotted curves).
2.1. Numerical integration of constitutive response
A possible approximation for the fractional diﬀerentiation of a function
y
ð
t
Þ
is (Oldham and Spanier,1974)
n
þ
1
ð
D
a
y
Þ ¼
1
ð
D
t
Þ
a
X
n j
¼
0
b
j
ð
a
Þ
n
þ
1
j
y
;
ð
6
Þ
where it is assumed that the spacing in time is uniform, i.e.,
n
y
¼
y
ð
n
D
t
Þ
. The coeﬃcients
b
j
ð
a
Þ
depend on theGamma function as follows:
b
j
ð
a
Þ ¼
C
ð
j
a
Þ
C
ð
a
Þ
C
ð
j
þ
1
Þ
:
ð
7
Þ
By using the recursion formula
C
ð
j
a
Þ
C
ð
j
þ
1
Þ ¼ ð
j
1
a
Þ
j
C
ð
j
1
a
Þ
C
ð
j
Þ
;
ð
8
Þ
it is possible to avoid the evaluation of the Gamma function; the coeﬃcients
b
j
ð
a
Þ
are given by
b
0
ð
a
Þ ¼
1
;
...
;
b
k
ð
a
Þ ¼ ð
k
1
a
Þ
k b
k
1
ð
a
Þ
;
...
k
¼
1
;
...
;
n
:
ð
9
Þ
For convenience, the expression in Eq. (6) can be rewritten as
n
þ
1
ð
D
a
y
Þ ¼
1
ð
D
t
Þ
a
ð
n
þ
1
y
n
y
Þ
;
ð
10
Þ
where
n
y
¼
X
n j
¼
1
b
j
ð
a
Þ
n
þ
1
j
y
ð
11
Þ
is a known quantity at time
t
n
þ
1
.
0 5 10 15Non dimensional time
t/
τ
1
1e061e041e021e+00
C r e e p r a t e d (
w / w
( t = 0 ) ) / d t
α=0.5α=0.3α=1.0
Fig. 4. Rate of creep displacement functions.
F. Barpi, S. Valente / International Journal of Solids and Structures 41 (2004) 2607–2621
2611