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The size effects in compression on drilled cylindrical concrete specimens obtained from a unique concrete block over a large scale range (1:19) are analyzed. The experimental results show scale effects on dissipated energy density rather than on the compressive strength. A theoretical explanation for such a phenomenon is presented, assuming a noninteger physical dimension of the sub-domain where dissipation occurs. A comparison between experimental and theoretical values is discussed and a renormalization procedure to obtain a scale independent constitutive law is presented. A scale-independent constitutive law in compression is put forward, which permits to define a unique relationship for softening in concrete. This goal is achieved by defining a fractal strain (or dilatation) whose fractal dimension is related to the sub-domain in which energy dissipation occurs.