Issue 38

M. Lutovinov et alii, Frattura ed Integrità Strutturale, 38 (2016) 237-243; DOI: 10.3221/IGF-ESIS.38.32 238 S ELECTED METHODS ue to the plane stress state at the notch tip, one component of principal stresses has a zero value and therefore, in case of principal components, there are five unknowns, for which five equations are needed. These unknowns are maximum principal strain ε 1 , middle principal strain ε 2 , minimum principal strain ε 3 , and two of principal stresses σ 1 , σ 2 , σ 3 depending on the type of loading. All of the selected methods, with the exception of Hoffmann–Seeger’s method, were defined in two different variants. The first estimate is obtained when Neuber’s rule is used for compiling the set of equations. The second estimate is obtained when the ESED rule [3], [6] is used instead. Hoffmann–Seeger’s method uses only Neuber’s rule. Both Neuber’s rule and the ESED rule state that strain energy density at the notch tip of a body with elastic–plastic behaviour is equal to strain energy density at the notch tip of a body with ideal elastic behaviour. The difference between these rules is that Neuber’s rule also takes in account complementary strain energy density [3]. As the first equation for prediction according to Moftakhar’s method, the extensions either of Neuber’s rule or the ESED rule are used. These extensions were developed for multiaxial loading and are represented by a sum of products of the individual stress and strain components. The next three equations for Moftakhar’s method are provided by Hencky’s equations that relate strains, stresses, and a material model represented by the cyclic stress–strain curve. The last equation is provided by the energy distribution assumption which states that fractional contribution of the largest principal notch tip stresses and strains to the total notch tip strain energy density [3] is equal for a body with elastic–plastic behaviour and a for a body with elastic behaviour. Besides the five unknown parameters mentioned above, there is another unknown which is a part of Hencky’s equations. This parameter is the plastic component of equivalent deformation. Moftakhar in [3] calculates it from bilinear approximation of cyclic stress–strain curve. Another way to calculate this parameter is by using the Ramberg–Osgood expression. Reinhardt’s method [4] represents a modification of Moftakhar’s method. The motivation for the modification was the fact that, since the original method forms a set of nonlinear algebraic equations, several solutions for that set exist in general [4]. Therefore, Reinhardt et al. presented a new algorithm for calculating notch tip stresses and strains based on the equations derived from the original Moftakhar’s set. In Hoffmann–Seeger’s method [5], another generalized form of Neuber’s rule is used for calculating actual equivalent stress. The method also uses Hencky’s expression, expression for von Mises stress, and an assumption about the ratio of surface strains to calculate the unknown components of stress and strain tensors. The selected methods designed for non-proportional loading work with general components of stresses and strains, and therefore there are seven unknowns ( ε 11 , ε 22 , ε 23 , ε 33 , σ 22 , σ 23 , σ 33 ), for which seven equations are needed. Since the ratio of deviatoric stresses in case of non-proportional loading is unlikely to be a constant value, the final stress–strain state is dependent on the applied loading path [6]. Therefore, the equations used for predictions should be in an incremental form. The set of equations for Singh’s method [6] consists of a) either generalized Neuber’s rule or the ESED rule, b) the Prandtl– Reuss relations, which reflect elastic–plastic material behaviour and provide four equations, and c) two equations provided by energy ration assumptions. The first three equations for Buczynski’s method [7] are obtained from either generalized Neuber’s or the ESED rule, expressed as an equality of symmetric strain energy density tensors. The remaining four equations are provided by the Prandtl–Reuss relation. V ERIFICATION PROCESS he verification process was as follows: at first the selected methods were implemented in two programs written in MATLAB, one for the case of proportional loading, the other one for non-proportional loading. In the next step, in order to verify whether individual methods are implemented correctly, the data from elastic simulations referred originally by the authors, were derived from the articles. After obtaining the original inputs, they were imported into the MATLAB programs. At the beginning of the estimation process, input vectors were divided by preselected increments. In case of methods designed for non-proportional loading, the end of the first increment was the yielding state. The yielding state was found by linear interpolation using von Mises stress (Eq. 1) for comparison with yield strength D T