Issue 38

R. Fincato et alii, Frattura ed Integrità Strutturale, 38 (2016) 231-236; DOI: 10.3221/IGF-ESIS.38.31 233 The mobility of the similarity centre is crucial in the model because it allows material ratcheting to occur during cycles, allowing the prediction of the plastic strain accumulation that is more reliable and realistic, and making the model suitable for fatigue investigation. A detailed explanation of the theoretical features is not the object of this paper and the reader is referred to Refs. [11] and [12] for a more detailed discussion. Damage In continuum damage mechanics, the damage variable is assumed to be an internal variable that includes the degradation of the mechanical performance arising from microscale imperfections and defects in the medium. Its evolution is associated with a dissipative mechanism derived from an elastic damage potential [4, 5, 13, 14], and the following assumptions are made. • The distribution of the defects inside the medium is uniform, which reduces the damage as a scalar isotropic variable. • Strain equivalence applies, where the strain behaviour is the same for damage or undamaged materials. • The effective stress, which includes the damage effect on the elastic response that is not included in the Gurson approach, is eff D . (1 )     σ σ σ (3) The coupling with the elastoplastic model is modified by Eq. (1) to include the damage variable, similar to Lemaitre [5], Benallal et al. [15], and De Souza et al. [2] as f F H f RF H ˆ( ) ( ); ( ) ( )     σ σ (4) In contrast to previous studies, the term is not associated with the stress function, ; therefore, the damage variable will not modify the original definition of the outward normal vector in the associated flow rule, simplifying the derivation of all the variables of the plasticity model. Without describing the details of the mathematical manipulations, the main variables are   p p p dF D c h h R F dH dF D M tr h U R c F dH R R 1 1 2 ˆ ˆ 1 ; ; 3 1 1 ˆ ˆ ˆ 1 1                                                                 σ σ s D s α s N σ σ σ N σ a D s σ        (5) where a is p o   a D , c is a material constant regulating the speed of the similarity centre, and U is a mathematical function for defining the similarity ratio rate according to [12]. The corotational stress rate can be written as a function of the total strain rate as p M tr              EN EN σ E D NEN (6) The dash over the elastic constant matrix, E , indicates that the elastic behaviour is affected by the damage. The damage evolution law is assumed to be s m m m and D f Y D H s Y s G K and D 2 2 2 3 2 2 1 0 0 0 ( ) ; ; 6 2 0 0 0                                             σ    (7)