NUMERICAL SIMULATION OF
CREEP & ALKALI AGGREGATE REATION and
TEMPERATURE
In CONCRETE DAMS
Based on Work of
Luiz Eloy Vaz
Total strain is decomposed as follows:
ε(t )  ε cr (t )  εT (t )  ε a (t )
(0.1)
Where ε cr , εT , and ε a correspond to creep, temperature and alkali-aggregate respectively.
Creep strain is modeled as
t
εcr (t )  H  F ( , t   )dσ( )
(0.2)
t0
Where
1
 

 
H





1







1

1


1

1   
and F ( , t ) is the creep function modeled by
t 
n 


1
 1 


F ( , t   ) 
 
1  e i   (0.3)
E0 ( ) j 1  E j ( ) 
 


E1 ,1
En , n
E0
1
where n is the number of elements in the generalized Kelvin chain,  i 
i
Ei
relaxation time
for damper i, and Ei ( ) Young’s modulus at time  .
Ei ( )  A1 
A2
1


3
A3
1

A4

90
1

2700
AAR Effect:
 ij
t

 0
 K log  ij
t
 0


  ij

(0.4)
 0
is the unconfined rate of strain increase,  0 the unconfined stress, K a factor yet
t
to be determined, and  ij the Kroenecker delta.
where
ε A (t )
d
t0

ε A (t )  
t
(0.5)
Temperature:
t
ε T   dε T
(0.6)
t0
SUBSTITUTE:
t
n
n t 
 1
   t  dσ 
t
t 
1 
1
A
ε(t )  H  

d
σ
(

)

H
exp
d


 d  t dεT  t



0
0 
E
(

)
E
(

)
E
(

)

d

j

1
j

1



j
j
j
t0 
t



 
0 
Using an incremental form for the previous discretization:
n
n
1
1 
ε iA1 i
i
i
ε  H  i   i  σ i   εf j  εT 
t
t
j 1
 E j 1 E j 
i
(0.7)
where E i is Young’s modulus inside t i and Eij is the average value of E j inside t i

   t  dσ 
1
εf j (t )    H
exp 
d
   d 
E
(

)
j
j
t0 




t
(0.8)
which can be rewritten as
E j (t )
Assuming that E j (t ) and
E j (t )
d
dσ
εf j (t ) 
εf j (t )  H
(0.9)
dt
j
dt
dσ
are constants inside the interval t i , the previous equation
dt
reduces to
εf j (t )  a   εf
i 1
j
   t  t i 1  

 a  exp 
  j

(0.10)
where
2
a
 σ i 
i 
 t 
Eij
 jH 
(0.11)
Considering Eq. (0.10) at time t i and subtracting εf ij 1 we obtain an expression for the
incremental deformation

 t i
εf ij  εf ij 1 exp  
 

j

  H ij σ i
(0.12)
  1 
i
E

j
 
where

 ij 
 t i  
 

j

 
 j 1  exp  

t i
(0.13)
From the previous two equations we determine the constitutive relation
σ i  Di  εi  εieff

(0.14)
where
Di  H 1
Ei
1   ij
1 E  i
j 1 E j
n
(0.15)
i
n

 t i  
i
i
εieff   εf ij 1  exp  
   εT  ε A (0.16)


j 1

j 



ε iA   Ai 1t i
(0.17)
t
 t i 1  i  2
  i 1 
εf ij 1  exp  
εf j  H  ji 1  σ i 1 (0.18)

  
E 
j 

 j 
Equations (0.14) to (0.18) enable us to characterize a linear elastic model with initial
deformation associated with time interval t i , hence through the superposition of strain
associated with creep, alkali-aggregate reaction and temperature we can effectively model
concrete aging.
3