PSI - Issue 23

A. Karolczuk et al. / Procedia Structural Integrity 23 (2019) 69–76 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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plastic strain energy densities are implemented, Ellyin (1997). In a general case, the function for multiaxial and time alternating stress and strain reduction on a critical plane can be presented as = ( ( ), ( ), , ) Ǥ (10) In many cases, Glinka et al. (1995), Łagoda et al. (1999), Pan et al. (1999), the proposed energy-based parameter does not directly represent the physical quantity of strain energy density. It is a function derived from the amplitudes of the stress and strain vector components on the critical plane. This kind of approach has gained in popularity. According to Liu (1993), Glinka et al. (1995), Chen et al. (1999), Pan et al. (1999), its effectiveness in correlating experimental and calculated fatigue lives is acceptable. The fact that the sum of the shear and normal strain energy densities has a physical background is an obvious advantage of energy-based criteria in comparison with stress- and strain-based parameters. However, the large number of proposed energy-based damage parameters and their progressive modifications suggest that the range of applicability of these energy models is limited to the given type of loading, material and other test conditions. One reason for the limited application of the proposed energy-based parameters is the assumption that the material parameters implemented in Eq. (10) are independent from the fatigue life. The energy-based parameter proposed by Glinka et al. (1995) and modified by Pan et al. (1999) is used below as an example in which the material parameters are derived as being life-dependent. 2.3.1. Example – Glinka et al. damage model Glinka et al. (1995) proposed the energy-based damage parameter as the sum of the shear stress amplitude , multiplied by shear strain amplitude , and of the product of the normal stress and strain amplitudes on the plane of maximum shear strain. In order to take into account the distinct effect of the shear and normal components on fatigue damage, two material parameters were introduced by Pan et al. (1999). The damage parameter takes the form: = , , + 1 2 , , (11) where the material parameters are 1 = ′ ′ ⁄ , 2 = ′ ′ ⁄ . The product of two material parameters can be replaced by one parameter, and the energy parameter that is obtained can be compared with a reference energy parameter, which is the product of the strain and stress fatigue characteristics, as follows , , + , , = ( ) ( ) Ǥ (12) The parameter is derived by applying the above damage parameter to the uniaxial alternate push-pull loading. Descriptions of the shear strain  f ( N f ) fatigue curve and the shear stress  f ( N f ) fatigue curve form the right-hand side of Eq. (12). In this case, the amplitudes of the strains and stresses occurring in Eq. (11) on the plane of the maximum shear strain are expressed as presented for the Matake and Fatemi-Socie models. The material parameter can be defined as life-dependent, and it is derived in the following form ( ) = 4 ′ ′ (2 ) 0 + 0 + 4 ′2 (2 ) 2 0 − 2 ′2 (2 ) 2 0 (1+ )−2 ′ ′ (2 ) + (1+ ) ′2 (2 ) 2 (1− )+ ′ ′ (2 ) + (1− ) Ǥ (13) 3. Experiment Hourglass bar specimens 10 mm in diameter in the critical section were used in an experimental study. The specimens were cut out from a rolled plate 38.1 mm in thickness made of 2124-T851 aluminium alloy. The measured roughness parameter R a after fine machining was below 0.50  m. The axis of the specimens coincided with the rolling direction. The dimensions of the bar specimens (minimum diameter 10 mm, no central cylindrical part, fillet radius to gripped ends 78 mm, gripped ends with a square cross-section of 12.9 mm on each side, 90 mm total length) enabled two specimens to be prepared within the plate thickness. The ultimate and yield tensile strengths in the rolling direction were found to be 477.1 MPa and 439.8 MPa, respectively. The specimens were subjected to fully reversed sinusoidal: (i) plane bending ( ⁄ = 0 ), (ii) torsion ( = ∞ ⁄ ), and (iii) two proportional combinations of bending and torsion loading ( ⁄ = 0 . 33 and 1.00) under force control. Details about the applied fatigue stand can be found in Karolczuk et al. (2015). Failure was