Issue 29

A. Caporale et alii, Frattura ed Integrità Strutturale, 29 (2014) 19-27; DOI: 10.3221/IGF-ESIS.29.03 20 compression. In this work, the micromechanical model presented in [17] is used in order to predict the failure surface of cement concrete subject to multi-axial compression. It is underlined that stress-strain curves in uni-axial compression are provided in [17], whereas failure curves in the plane of the principal stresses in concrete are considered in this work. The failure curves provided by the proposed micromechanical method can also be obtained with experimental tests which involve high economical costs as equipment able to impose bi-axial and tri-axial states of compression is required. The aim of the work is to determine theoretically the mechanical parameters that affect more the behavior of cement concrete in order to reduce the number of expensive experimental tests on concrete specimens. M ICROMECHANICAL MODEL n the proposed model, the concrete material is viewed as a composite with the following constituents: coarse aggregate (gravel), fine aggregate (sand) and cement paste. The cement paste contains some voids which grow during the loading process. In fact, the non-linear behavior of the concrete is attributed to the creation of cracks in the cement paste; the effect of the cracks is taken into account by introducing equivalent voids (inclusions with zero stiffness) in the cement paste. The three types of inclusions (namely gravel, sand and voids) have different scales, so that the overall behavior of the concrete is obtained by the composition of three different homogenizations; in the sense that the concrete is regarded as the homogenized material of the two-phase composite constituted of the gravel and the mortar; in turn, the mortar is the homogenized material of the two-phase composite constituted of the sand inclusions and a (porous) cement paste matrix; finally, the (porous) cement paste is the homogenized material of the two-phase composite constituted of voids and the pure paste; the pure paste represents the cement paste before the loading process, so that it does not contain voids or other defects due to the loading process. The pure paste can contain defects but these are due to the production process of the cement paste and not to the loading process. The above-mentioned three homogenizations are realized with the predictive scheme of Mori-Tanaka in conjunction with the Eshelby method, frequently used in the homogenization of composites [11,13]. The micromechanical method described in [17] provides the stress in the concrete material subject to a prescribed strain and, vice versa, the strain in concrete material subject to a prescribed stress. The volume fraction of the voids in the cement paste is denoted by v f . The overall elasticity of the porous cement paste, mortar and concrete are denoted by   p C ,   m C and   c C , respectively. Considering that the paste is composed of pure paste and voids, the overall elasticity   p C of the paste is a function of the volume fraction v f of the voids. As a consequence, the overall elasticity of mortar and concrete are also functions of v f . The overall elasticity of paste, mortar and concrete are evaluated with the Mori- Tanaka method in conjunction with the result of the Eshelby’s problem of an ellipsoidal inclusion in a homogeneous, linearly elastic and infinitely extended medium (see also [17]). The overall compliances of the cement paste, mortar and concrete are denoted by   p D ,   m D and   c D , respectively; these compliances are obtained by inverting the corresponding overall elasticity and are also functions of v f :                               1 1 1 p p v v m m v v c c v v f f f f f f       D C D C D C (1) In this work, a macro-stress  σ is prescribed to the concrete and the average stress and strain in the constituents of concrete and corresponding to  σ are evaluated with the following iterative procedure, which makes use of a secant approach. The value of the void volume fraction v f at the generic i th iteration is denoted by , v i f . The average stress and strain in the constituents of concrete at the generic i th iteration are evaluated by assuming that it is known  , 1 v i f , i.e. the void volume fraction at the previous iteration  1 i . Specifically, the following overall elasticity and compliance corresponding to  , 1 v i f are evaluated in the i th iteration of the procedure:                               , 1 , 1 , 1 , 1 , 1 , 1 , , , , , p p m c m c v i v i v i v i v i v i f f f f f f C C C D D D (2) I