Issue 37

C. Madrigal et alii, Frattura ed Integrità Strutturale, 37 (2016) 8-14 DOI: 10.3221/IGF-ESIS.37.02 8 Focussed on Multiaxial Fatigue and Fracture Plastic flow equations for the local strain approach in the multiaxial case C. Madrigal, A. Navarro, C. Vallellano Dpto. Ing. Mecánica y Fabricación, Escuela Técnica Superior de Ingeniería Avda. Camino de los Descubrimientos, s/n. 41092, Seville University of Seville navarro@us.es A BSTRACT . This paper presents a system of plastic flow equations which uses and generalizes to the multiaxial case a number of concepts commonly employed in the so-called Local Strain Approach to low cycle fatigue. Everything is built upon the idea of distance between stress points. It is believed that this will ease the generalization to the multiaxial case of the intuitive methods used in low cycle fatigue calculations, based on hysteresis loops, Ramberg ‐ Osgood equations, Neuber or ESED rule, etc. It is proposed that the stress space is endowed with a quadratic metric whose structure is embedded in the yield criterion. Considerations of initial isotropy of the material and of the null influence of the hydrostatic stress upon yielding leads to the realization of the simplest metric, which is associated with the von Mises yield criterion. The use of the strain ‐ hardening hypothesis leads in natural way to a normal flow rule and this establishes a linear relationship between the plastic strain increment and the stress increment. K EYWORDS . Low cycle fatigue; Plastic Flow Rule; Kinematic Hardening; Non-proportional Loading; Multiaxial Fatigue I NTRODUCTION e are trying to develop a theory of cyclic plasticity which allows fatigue designers to make calculations for multiaxial loads in a way as similar as possible to which they do when using the well-known Local Strain methodology for uniaxial low cycle fatigue problems. We would like to define concepts that translate to multiaxial loadings in a simple manner the tools of that trade, namely, the use of the Cyclic Stress-Strain Curve and hysteresis loops, the invocation of the memory rule when hysteresis loops are “closed”, the extension of the Neuber or ESED rules to multiaxial loading, etc. We have found it useful to base our theory in the idea of distance between stress points and to calculate these distances by using the expression for the yield criterion [1-6]. The Local Strain Method constitutes nowadays a standard tool for fatigue life predictions in many industries. It has been incorporated in commercial software [7, 8] and it is very well described in textbooks [9, 10]. The extension of the Local Strain Method to the multiaxial case requires at least three main steps. The first one is the development of plastic flow rules which reproduce the way we operate with hysteresis loops, cyclic curves, memory effect and so on in the simple uniaxial case. The second step would be the development of multiaxial Neuber-type rules for dealing with inelastic strains at notches. This relies heavily on the use of a theory of plasticity and hence on the previous step. There are already a W