Relaxation and relief in stressed solids in terms of superellipse equation: concrete

Abstract

There is a large group of monotonic exo-processes, such as relaxation, shrinkage and maturity, which is realised over finite time and whose first derivative is infinite at the beginning of the process and zero at the end. The differential equation of exo-processes is given in this paper. Its analytical solution represents the left-hand upper quadrant of a superellipse and contains the time-operator, which links the exo-process with time. It is shown that relaxation can be successfully described by an exo-equation without recourse to viscosity, plasticity, rheological models or regression equations. The analytical expression for the operator of relaxation comprises three material parameters – the duration of total relaxation, the actual potential of relaxation and the exponent of the superellipse. The half-realisation factor, which reflects the rate of relaxation, is defined. The operator of relaxation is independent of the stressing, being a transparent function of time. It correctly represents the stress drop even for a fraction of a second, a fact of the highest importance in the context of fatigue. The superellipse equation describes relaxation in a wide range of solids besides concrete, giving clear evidence that it is a distinct process with its own physics and rules. The operator of instant relaxation, called relief (usually taken as a decrease in the apparent elastic modulus) is obtained from the equation of relaxation, where the time under load is represented by the reciprocal of the rate of loading. The operator of relief is independent of the stresses and is constant for a fixed loading rate. For structural concrete under a normative rate of loading, relief can account for 18% of the decline in the static versus dynamic modulus of elasticity.