“Computational Civil Engineering - "Intersections" International Journal

120 S. Oller, Al.H. BarbatANNEX: CONTINUUM CONSTITUTIVE DAMAGE LAWIntroduction to isotropic damage modelThis annex contains a brief review of the isotropic continuum damage model at apoint of a structure [19], which is used in the paper to formulate the damage of thecross section of a bridge pier. The damage at a point of a continuous solid isdefined as the degradation of the stiffness and strength due to the decrease of theeffective area [11]. The continuum theory of the damage was formulated byKachanov [12] in the creep behavior context, but later on it has been reformulatedand accepted as a valid alternative to simulate the rate independent behavior ofseveral materials [4-6, 14-17, 26, 27].Formulation of isotropic damage modelDegradation of the material properties happens due to the presence and growth ofsmall cracks and voids inside the structure of the material. This phenomenon canbe simulated by means of the continuum mechanics taking into account a scalar otensorial internal damage variable. This internal variable of damage measures thelevel of degradation of the material in a point and its evaluation is based on thetransformation of the real stresses in other effective stresses. For the simpleisotropic damage used here, the relationship between the real and the effectivestress is described using an isotropic damage variable dσσ 0 =(A.1)(1 − d )In this equation, d is the internal variable of damage; σ it is the Cauchy stresstensor and σ 0 is the effective stress tensor, evaluated in the “no-damaged” space.This internal variable represents the loss of stiffness level in a point of the materialand its upper and lower limits are given by0 ≤ d ≤1(A.2)The upper limit (d=1) represents the maximum damage in a point and the lowerlimit (d=0) represents a non damaged point.Helmholtz free energy and constitutive equationThe Helmholtz [18] free energy for the isotropic damage model is given by theexpression