Engineering approaches to multiaxial and non‐proportional fatigue of notched components

Abstract

Many components in engineering applications are subjected to multiple and uncorrelated loads during service‐life. Thus, multiaxial stress states with rotating principal axis may occur. For this special case of multiaxial and non‐proportional stresses the results of many fatigue assessment methods which are used in the industrial practice are of poor quality. Fatigue lifetimes of shoulder shafts (quenched and tempered steel) are estimated on the basis of the extended short crack model in combination with a multiaxial notch approximation. This approach shows a high accuracy but the precise modelling of non‐proportional hardening effects requires a complex plasticity model. Therefore, a simplified approach for considering non‐proportional hardening is introduced. Thus, the calculation method gets applicable in the engineering practice. Results are compared to well‐established engineering approaches. Furthermore, new component tests on die‐cast housings with two load channels under constant and variable amplitude loading are presented and discussed. The loads are applied in‐phase (proportional) as well as out‐of‐phase, which results in a high non‐proportional stress path at the crack initiation site. The effects of multiaxial and non‐proportional stress states seem to play a minor role in the fatigue assessment of die‐cast housings.

On Scaled Normal Stresses in Multiaxial Fatigue and Their Exemplary Application to Ductile Cast Iron

“…σ eq,signed = sign(I 1 )•σ eq = sign(σ 1 +σ 2 +σ 3 )•σ eq (10) Contrary to the frequently encountered definition of the sign function, where sign(0)= 0 (11) here sign(0)= 1 (12) is assigned since, otherwise, due to the sign function, an equivalent stress of 0 would result for stress states that clearly represent a loaded component, e.g., pure torsion with σ 2 = −σ 1 and σ 3 = 0. Many applications show a plane stress state in the critical area if the latter is located at the component surface and if the surface is not loaded by pressure or friction forces.…”

“…Equivalent stress-time series can easily be calculated for proportional stress-time series by using this procedure. However, as sufficiently known from the literature, problems arise when applying it to nonproportional stresses, which can be easily illustrated by two examples [11,36,37], shown in Figure 5: • Example 1, Figure 5, left side: A dominant shear stress amplitude is superimposed with a high-frequency normal stress of small amplitude (Figure 5a). The corresponding principal stresses and course of the first invariant that determines the later determined sign are shown in Figure 5b.…”

“…One approach that can meet the described requirements for the calculation of fatigue strength under nonproportional loading is the critical plane approach with scaled normal stresses. It was originally proposed by Gaier and Dannbauer [9,10] and was later adopted in modified form by Riess et al [11,12]. One of the major issues in dealing with nonproportional loading is the occurrence of two effects:…”

The contrast of simplicity and accuracy in modeling multiaxial notch fatigue

“…A first approach to scale the normal stress in critical plane was introduced by Gaier and Dannbauer [9]. A similar, but more efficient method was recently proposed by Riess et al [10]. The multiaxial effect is modeled depending on the current extent of multiaxiality.…”

“…(2) is given in section 3.2.2. This function is the result of a simple interpolation between normal stress hypothesis and distortion energy hypothesis on basis of the shear strength ratio f w,τ = τ w /σ w [10]. Fig.…”