Effect of the Transition Zone on the Bulk Modulus of Concrete
- PDF / 1,048,197 Bytes
- 6 Pages / 414.72 x 648 pts Page_size
- 23 Downloads / 187 Views
Mat. Res. Soc. Symp. Proc. Vol. 370 01995 Materials Research Society
inclusions are spherical is justified by the fact that slight variations from sphericity are of minor importance, particularly if the inclusions are stiffer than the matrix [7]. We then utilize the closed-form analytical solution for the effective bulk modulus of this type of material that has recently been found by Lutz and Zimmerman [8]. The predicted effective bulk modulus depends on "known" parameters such as the volume fraction of sand and the elastic moduli of the sand and the bulk cement paste, as well as on the elastic moduli at the interface with the inclusions. If the model is used to "invert" measured data, the elastic moduli at the interface can be inferred. This in effect yields a quantitative and non-destructive means of estimating the properties of the interfacial transition zone. MODEL OF THE TRANSITION ZONE AS A RADIALLY-INHOMOGENEOUS REGION We propose a conceptual model of concrete as shown in Fig. 1. The aggregate particles are assumed to be spherical, with radius a. Outside of each inclusion, the two elastic moduli vary smoothly with radius, and approach those of the bulk cement paste as r increases. The precise manner of variation of the moduli will not generally be known, and the assumption of a smooth variation is of course an approximation. The main requirements of the equations used to model the variation in moduli are that they decay away from the interface, and asymptotically level off to some constant values that represents the bulk cement paste. Furthermore, for the model to be sufficiently general and flexible, the values of the elastic moduli at the interface, as well as the thickness of the interphase zone, should be adjustable parameters. With these conditions in mind, and guided in part by the findings of Theocaris [9], we will assume that the two moduli K and G vary according to K(r) = Krp +(Kif -KcP)(r/a )- 0,
(1)
G (r) = Gp + (Gif - Gp)(r/a)-0,
(2)
Fig. 1. Schematic diagram of the elastic moduli variation described by eqs. (1,2), superimposed over a photograph of an interfacial transition zone. 414
where a is the radius of the inclusion, the subscript cp refers to the bulk cement paste, and the subscript if refers to the interface with the inclusion. The parameter f3controls the rate at which the moduli decay away from the inclusion; larger values of P3 correspond to interphase zones that are more localized. In order for the analytical solution found by Lutz and Zimmerman [81 to be applicable, 03must be an integer. As the moduli variations will never be known precisely, this restriction poses no practical limitation on the applicability of the model. Finally, note that in solving the elasticity equations, it is convenient to use the moduli K and G instead of the commonly-used engineering parameters E and v. However, these parameters are related to each other through the identities 9KG 3K+G'
3K-2G 6K+2G
(3)
Lutz and Zimmerman [8] found a closed-form solution for the displacements, stresses and strains
Data Loading...
