Page 111 - Handbook of Materials Failure Analysis

106 CHAPTER 5 Failure analysis of reinforced concrete structures

reinforcements. It is worth to mention that all influence of the improvements on shear
representation is accounted during the nonlinear procedure.

5 FICK’S DIFFUSION LAW
Corrosion of reinforcements induced by chlorides occurs in the presence of oxygen
and moisture when the chloride build-up within the structures exceeds a threshold
value. Even for carefully constructed concrete structures, with negligible or practi-
cally non-chloride inherited at the construction stage, the gradual build-up of chlo-
ride content takes place slowly through ingress of chlorides from external sources.
The transport phenomenon associated with the movement of chlorides along
structures exposed to aggressive environments is attributed, in most part, to diffusion
of chloride ions into concrete pores under a concentration gradient. The coefficient of
chloride diffusion, which depends upon the pore structure of concrete, characterizes
such flow under a given external concentration of chloride. This parameter is
assumed as a characteristic property of hardened concrete.
To simulate the chloride ingress and its transport into concrete pores, Fick’s dif-
fusion law [43] has been widely considered as an acceptable model
[2,9,15,17,20,32,34,35]. Fick’s laws for diffusion are applicable for homogeneous,
isotropic, and inert materials [44]. Moreover, the mechanical properties related to
diffusion process are assumed to be identical along all directions and kept constants
along time. Considering concrete, these hypotheses are not completely satisfied,
because concrete is well known as heterogeneous, anisotropic, and chemically reac-
tive (continued hydration and microcracking process) material. However, the
methods commonly adopted for chlorides transportation modeling in concrete con-
sider this process governed only by ionic diffusion. Then, it assumes that the concrete
cover is completely saturated. Therefore, it makes the hypotheses of Fick’s law
acceptable for the chloride ingress modeling, because, in this case, the material is
assumed completely saturated, with unidirectional chloride flux, that is, from the
exterior surface into the concrete depth. When chloride diffuses into concrete, a
change in chloride concentration, C, occurs at any time, t, in every point, x, of the
concrete, that is, it is a non-steady state of diffusion. To simplify its analysis, the dif-
fusion problem is considered as one-dimensional. Many engineering problems of
chloride ingress, as those discussed in this study, can be solved considering this
simplification.
Fick’s diffusion theory assumes that the transport of chlorides into concrete
though a unit section area of concrete per unit of time (the flux F) is proportional
to the concentration gradient of chlorides measured at normal direction of section.
Then:
@C
F ¼ D c (5.28)
@x
The negative sign on the equation above arises because the diffusion of chlorides
occurs in the opposite direction of the increasing concentration of chlorides. The