Issue 29

A. Caporale et alii, Frattura ed Integrità Strutturale, 29 (2014) 19-27; DOI: 10.3221/IGF-ESIS.29.03 23 between 30 and 50,  is equal to 0.8,  , ,0 v f and , v m f are assumed equal to zero. In the green curves of Fig. 1, ,0 v f is assumed equal to zero, , v m f varies between 10 and 20, , v eq f is equal to 30,  and  are assumed equal to 0.8. The tangent line to a curve of Fig. 1 at the origin of the  11 - and  11 -axes is the overall initial stiffness 0 E . Each curve of Fig. 1 has a maximum denoted by  p and attained when    11 p . The tangent line to the curves of Fig. 1 at the end- point     , p p is about horizontal. In case of uni-axial compression, the peak  p of the stress-strain curve represents the compressive strength  c of concrete. The curves of Fig. 1 have the same initial stiffness 0 E : in fact, 0 E depends on the properties reported in Tab. 1 and the initial porosity ,0 v f , which are constants that do not vary among the curves of Fig. 1. The curves with a given value of  have the same ratio   p p : e.g. in the black curves characterized by   0.7 , the ratio   p p is about 0 2 E ; in the red and green curves characterized by   0.8 , the ratio   p p is always constant and is greater than 0 2 E . The exponent   0.7 was chosen in order to have    0 2 p p E , as predicted by the following phenomenological curve proposed by Desayi and Krishnan [19]            0 11 11 11 2 11 1 p E (9) The generic curve of Fig. 1 is obtained by connecting the point     11 11 , ; the end-point of this curve is     , p p , whose coordinates are used to have the dimensionless curve       11 11 , p p . In Fig. 2,   11 p is plotted against   11 p for the eight curves of Fig. 1. The eight dimensionless curves of Fig. 2 are the same; this behavior is also exhibited by the experimental stress-strain curves and the curve (9). Constituent Young’s modulus Poisson’s ratio Volume fractions in cement concrete Gravel 45 GPa 0.23 0.4 Sand 65 GPa 0.21 0.326 Cement paste - - 0.274 Pure paste 15 GPa 0.22 - Table 1 : Geometrical and mechanical properties of the four-phase composite. 0,000 0,020 0,040 0,060 0,080 0,100 0,120 0,000 0,001 0,002 0,003 0,004 0,005 0,006   (GPa)   f v,eq , f v,m    = 0.8, f v,eq = 30, f v,m = 10, 20    = 0.7, f v,m = 0, f v,eq = 30, 40, 50    = 0.8, f v,m = 0, f v,eq = 30, 40, 50 Figure 1 : Stress-strain curves of concrete in uni-axial compression.