We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

We consider a class of probability measures µ α s,r which have explicit Cauchy-Stieltjes transforms. This class includes a symmetric beta distribution, a free Poisson law and some beta distributions as special cases. Also, we identify µ α s,2 as a free compound Poisson law with Lévy measure a monotone α-stable law. This implies the free infinite divisibility of µ α s,2 . Moreover, when symmetric or positive, µ α s,2 has a representation as the free multiplication of a free Poisson law and a monotone α-stable law. We also investigate the free infinite divisibility of µ α s,r for r = 2. Special cases include the beta distributions B(1 − 1 r , 1 + 1 r ) which are freely infinitely divisible if and only if 1 ≤ r ≤ 2.

This paper describes the elastic compliance matrix and Young’s modulus of directionally solidified (DS) plates which are composed of five columnar grains. Threedimensional finite element method analyses (FEM), taking account of the anisotropy of a crystal, were made to determine the elastic compliance matrix of the DS plate. The elastic compliance matrix calculated in the FEM analysis was reduced to an orthotropic matrix after giving an appropriate rotation of coordinates. The elastic compliance matrix calculated by Reuss and Voigt averages were compared with the FEM results. The two averages precisely estimated the FEM results but the Reuss model had a better accuracy. The Young’s modulus in any direction was discussed in both the coaxis and non-coaxis models.

This paper describes studies of the tension-torsion multiaxial low cycle fatigue of CMSX-2 Ni-base single crystal superalloy. Tension-torsion low cycle fatigue tests were carried out at 1173K using CMSX-2 hollow cylinder specimens aligned with the {001} axis. Several multiaxial strain and stress parameters were applied to the experimental data to examine the suitability of the parameters to life prediction. All the strain parameters proposed so far gave a large scatter of the data correlation. Discrepant data correlation with the strain parameters resulted from the anisotropic stress response due to the crystallographic texture. Larger Mises equivalent stress was applied in torsion tests than in tension tests at the same Mises strain. However, Mises stress and the equivalent stress based on crack opening displacement gave a satisfactory data correlation. This study developed a new equivalent strain, taking account of the anisotropy of the elastic constants, which correlates the multiaxial low cycle fatigue data with a small scatter.

This paper investigates the effect of solidification direction on cyclic constitutive relation and low cycle fatigue life for a directionally solidified (DS) superalloy Mar-M247LC at 1173 K in air. Strain controlled push-pull low cycle fatigue tests were carried out for three kinds of specimens of which loading direction was 0, 45 and 90 deg to DS axis. The 0 deg specimen exhibited the largest fatigue strength, the 45 deg specimen the smallest and the 90 deg specimen the intermediate. The difference in the fatigue strength due to the loading direction was discussed in detail from the viewpoint of the anisotropy of Young’s modulus. A life prediction method for DS specimens, considering the anisotropy of the Young’s modulus, was proposed and successfully applied to the low cycle fatigue data.