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A fractional order rate approach for modeling concrete structures subjected to creep and fracture

A fractional order rate approach for modeling concrete structures subjected to creep and fracture
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  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 quasi-static fracture. The attention is focused on theinteraction between strain-softening and time-dependent behaviour: a viscous rheological element (based on a frac-tional 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 three-point bending conditions (three different load levels) shows a good agreement both interms of failure-lifetime, and, load–displacement.   2004 Elsevier Ltd. All rights reserved. Keywords:  Cohesive; Concrete; Crack; Creep; Fracture; Fractional; Long-term behaviour; Softening; Viscosity 1. Introduction The long-term 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 mi-crocracking increases and stresses decrease as the overall deformation increases. This results in the soft-ening 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 evolu-tionary 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 strain-softening and time-dependent behaviour is analysed, with the emphasison very slow or quasi-static 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.: +39-11-5644886; fax: +39-11-5644899. E-mail addresses: (F. Barpi), (S. Valente). 1 Tel.: +39-11-5644853; fax: +39-11-5644899.0020-7683/$ - 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–  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 ðÞ ¼  d a ðÞ 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 Þð  x þ n Þ    y  ð t  Þ  generic function of time U 1  a ð t  Þ  kernel of the non-integer differentiation definition b i ð a Þ  i th coefficient of the numerical approximation of the non-integer 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 rate-dependent softening that has been developed in a series of paperby Bazant and co-workers (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 time-dependent 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 three-point bending tests described by Zhou (1992). 2. Description of the rheological model Rheology is concerned with time-dependent 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 well-known 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    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    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 non-integer 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 non-integer derivative(dashed and dashed-dotted 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 1e-061e-041e-021e+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
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