Issue 38

M. de Freitas et alii, Frattura ed Integrità Strutturale, 38 (2016) 121-127; DOI: 10.3221/IGF-ESIS.38.16 124 b c ESWT N 2 2 max ( ) max ( ) (2 ) 2                   (7) This equation can be simplified in the studied proportional history (which has zero mean stresses), because in this fully- alternate case the peak normal stress   max (  ) perpendicular to a Case A plane along θ is equal to the normal amplitude   (  )/2 . Therefore, the ESWT equation becomes Wöhler’s curve using Basquin’s formulation: b b c c N N 2 2 2 max ( ) ( ) ( ) max ( ) max (2 ) max (2 ) 2 2 2                                                    (8) Thus, the damage parameter to be maximized in the ESWT model simply becomes a a 2 ( ) / 2 | cos sin 2 |            for a fully-alternate cyclic loading. Deriving this expression and equating it to zero, the critical-plane angle θ ESWT with respect to the x -axis, represented in the first quadrant 0º ≤ θ ≤ 90º , is obtained from a a a ESWT p a 2 2cos sin 2 cos 2 0 tan 2 tan 2 2 tan                   (9) where λ is the SAR from the SSF model [6], and θ p is the principal direction from the first quadrant. Not surprisingly, θ ESWT is one of the fixed principal directions θ p of such proportional tension-torsion loadings. On this principal plane, the damage parameter is maximized, resulting in the principal stress equation for the normal and shear amplitudes σ a and τ a : a a a ESWT ESWT a a ESWT 2 2 4 ( ) 1 cos 2 sin 2 2 2 2                  (10) From the definition λ  tan  1 (τ a /σ a ) , it follows that τ a  σ a  tan(λ) ; thus this maximized damage parameter on the θ ESWT plane can be expressed as a function of σ a and λ : a a a ESWT a 2 2 2 2 4 tan ( ) 0.5 0.25 tan 2 2                        (11) Assuming ESWT’s model is able to predict crack initiation, the above expression would explain why the SSF can be represented as a function of σ a and λ , however requiring a fifth-order polynomial function to approximately reproduce such a non-linear expression. Finally, from the ESWT equation it follows that the predicted fatigue life N ESWT (in cycles) is   b b b a a a ESWT a ESWT c c c N 1/ 1/ 1/ 2 2 2 4 ( ) 0.5 0.5 0.5 0.5 0.25 tan 2 2                                            (12) R ESULTS AND DISCUSSIONS Material and loading paths he reasoning derived in the previous sections is applied to the experimental results published by Anes et al. [6] for proportional tension-torsion multiaxial fatigue tests carried out in low-alloy steel 42CrMo4, heat-treated by austenitizing, quenching, and tempering to improve its mechanical properties. The chemical composition and the monotonic and cyclic properties of 42CrMo4 are available in [6, 7]. Fatigue tests are carried out through a servo-hydraulic tension-torsion machine under stress control at room temperature, applying proportional loading paths. In order to perform the SSF mapping, five different proportional loading paths are T