Differential equation of concrete behaviour under uniaxial short-term compression in terms of atrophy (degeneration) and its solution

The Paper deals with the problem of the differential equation of concrete behaviour under load in the non linear stage of the stress-strain curve (SSC). The approach is based on following principles: the nonlinear behaviour of concrete is a consequence of the atrophying process of microcracking in it, because of its heterogeneity; the intrinsic elastic modulus ought to be taken constant up to peak stress; the specific stresses remain linearly related to the strains, also up to peak stress. The nonlinear behaviour of plain concrete both up to and beyond the strength limit is described by differential equations, which relate the response of the concrete to the strains in terms of atrophy (degradation) and the specific stresses in it, and represent SSCs for material affected by microcracking but without initial cracks comparable in size to the specimen dimensions. The obtained primary equation represents the total differential of concrete survival as a function of the strains in it up to the peak stress, and its solution is independent of the pattern of the survival function. For the descending branch the primary equation has to be supplemented by the confinement factor and by a function of macrodestruction. Solving this equation, a descriptive model for the nonlinear domain of the ascending branch of the SSC, called the central function, was obtained, using a gaussian survival function. It gives the relationship between the nominal stresses and the strains and atrophy of concrete based on the main parameters: intrinsic elastic modulus E, atrophy threshold ε_a, peak strain and the scattering factor d. A redefinition of strength in terms of the limiting atrophy and the strength formula is given, based on the balance between the energy added to the live part of the cross-section and the energy loss due to the increment of concrete atrophy. It is shown that the peak point strain ε_p not the primary parameter but is a function of ε_a, and d. The total similarity of the SSC for concrete and rock can be interpreted as proof of the validity of the proposed differential equations for brittle heterogeneous solids.