Mårten Olsson

Applying multiaxial fatigue criteria to standing contact fatigue

Abstract The ability of some multiaxial fatigue criteria to predict initiation of standing contact fatigue cracks is investigated. In the standing contact fatigue test an indenter subjects a case-hardened test specimen to a stationary but pulsating contact load. The initiation of two axi-symmetric crack types appearing in the test specimen is investigated. The surface initiated ring/cone cracks circumscribe the contact area, whereas the lateral cracks are horizontal sub-surface cracks. The effect of the stress-state history at possible crack locations is evaluated through the Sines, Haigh principal stress, Findley, Mc Diarmid and Dang Van multiaxial fatigue criteria. The material fatigue parameters of each criterion are determined from independent bending and torsion fatigue testing. Finally, the mean and spread in radial position of the ring/cone crack are evaluated by considering the statistical effects of a weakest link assumption using a three parameter Weibull distribution. The investigation shows that it is hard to distinguish a single criterion that well describes all aspects of the experimental results. For the current contact situation with highly compressive mean stresses in combination with tensile maximum values the Findley criterion shows the best overall performance followed by the Haigh principal stress criterion.

Tooth Interior Fatigue Fracture — computational and material aspects

The hypothesis of the gear failure mode Tooth Interior Fatigue Fracture (TIFF), i.e. a fatigue crack is initiated in the interior of the tooth, is analysed by numerical simulations using FEM and the critical plane fatigue initiation criterion according to Findley. The residual stress profile in the gear is simulated and calibrated versus a neutron diffraction residual stress measurement. Fatigue properties are determined experimentally and are taken to vary with depth in the numerical analysis. In addition, the effect of shot peening is included. The analysis shows that a TIFF-crack is initiated approximately mid-height of the tooth and slightly below the case-core boundary. Furthermore, the analysis shows that TIFF is more pronounced in idlers than in single stage gears.

Initiation and growth of standing contact fatigue cracks

Abstract In the standing contact fatigue test an indenter subjects a case hardened test specimen to a stationary but pulsating contact load. Two crack types appearing in the test specimen are investigated. The ring/cone cracks are surface cracks that circumvent the contact area, whereas the lateral cracks are horizontal sub-surface cracks. The initial crack lengths are determined for both crack types. Actual crack paths from experiments are evaluated numerically. For each crack tip position, stress intensity factors are determined from J1 and J2 integrals. The stress intensity ranges are compared to criteria for fatigue crack propagation rate and direction. As the cracks propagate, they orient in the direction with mode II loading close to zero.

Analysis of crack propagation during tooth interior fatigue fracture

Tooth interior fatigue fracture is a failure mode that is initiated as a fatigue crack in the interior of the tooth of a gear. TIFF cracks have been observed in case hardened idler gears. A fracture mechanical analysis of a TIFF crack is performed utilising FEA. A 3D TIFF crack is modelled at a position in the tooth that corresponds with an observed crack surface. The different material properties in the case and the core, determined by mechanical testing, are considered, as well as the residual state of stress due to case hardening. Various crack lengths are analysed to estimate crack propagation both into the core and into the case. The following results have been found: A TIFF crack initiated slightly under the case layer will propagate into the case layer where it stops. . The main crack propagation will take place in the core. . The crack propagation is only a small portion of the total life (order of 10(5) cycles). . After reaching the case layer the TIFF crack eventually deflects toward the tooth root and the upper part of the tooth falls off. The crack deflection is due to redistribution of contact loading. Several gear teeth pairs are simultaneously in contact and the cracked tooth is loaded less than the uncracked during this stage of life.

A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions

In the second-order reliability method (SORM), the probability of failure is computed for an arbitrary performance function in arbitrarily distributed random variables. This probability is approximated by the probability of failure computed using a general quadratic fit made at the most probable point (MPP). However, an easy-to-use, accurate, and efficient closed-form expression for the probability content of the general quadratic surface in normalized standard variables has not yet been presented. Instead, the most commonly used SORM approaches start with a relatively complicated rotational transformation. Thereafter, the last row and column of the rotationally transformed Hessian are neglected in the computation of the probability. This is equivalent to approximating the probability content of the general quadratic surface by the probability content of a hyperparabola in a rotationally transformed space. The error made by this approximation may introduce unknown inaccuracies. Furthermore, the most commonly used closed-form expressions have one or more of the following drawbacks: They neither do work well for small curvatures at the MPP and/or large number of random variables nor do they work well for negative or strongly uneven curvatures at the MPP. The expressions may even present singularities. The purpose of this work is to present a simple, efficient, and accurate closed-form expression for the probability of failure, which does not neglect any component of the Hessian and does not necessitate the rotational transformation performed in the most common SORM approaches. Furthermore, when applied to industrial examples where quadratic response surfaces of the real performance functions are used, the proposed formulas can be applied directly to compute the probability of failure without locating the MPP, as opposed to the other first-order reliability method (FORM) and the other SORM approaches. The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely used SORM approaches, an empirical SORM formula as well as FORM, are compared to the proposed method with regards to accuracy and computational efficiency. All methods have also been compared when applied to a wide range of hyperparabolic limit-state functions as well as to general quadratic limit-state functions in the rotationally transformed space, in order to quantify the error made by the approximation of the Hessian indicated above. In general, the presented method was the most accurate for almost all studied curvatures and number of random variables.

Reliability Based Design Optimization Using a Single Constraint Approximation Point

Cost and quality are key properties of a product, possibly even the two most important. Onedefinition of quality is fitness for purpose. Load-bearing products, i.e. structural components,loose their fitness for purpose if they fail. Thus, the ability to withstand failure is a fundamentalmeasure of quality for structural components. Reliability based design optimization(RBDO) is an approach for development of structural components which aims to minimizethe cost while constraining the probability of failure. However, the computational effort ofan RBDO applied to large-scale engineering problems has prohibited it from employment inindustrial applications. This thesis presents methods for computationally efficient RBDO.A review of the work presented on RBDO algorithms reveals that three constituentsof an RBDO algorithm has rendered significant attention; i ) the solution strategy for andnumerical treatment of the probabilistic constraints, ii ) the surrogate model, and iii) theexperiment design. A surrogate model is ”a model of a model”, i.e. a computationally cheapapproximation of a physics-based but computationally expensive computer model. It is fittedto responses from the physics-motivated model obtained via a thought-through combinationof experiments called an experiment design.In Paper A, the general algorithm for RBDO employed in this work, including the sequentialapproximation procedure used to treat the probabilistic constraints, is laid out. A singleconstraint approximation point (CAP) is used to save computational effort with acceptablelosses in accuracy. The approach is used to optimize a truck component and incorporatesthe effect that production related design variables like machining and shot peening have onfatigue life.The focus in Paper B is on experiment design. An algorithm employed to construct anovel experiment design for problems with multiple constraints is presented. It is based onan initial screening and uses the specific problem structure to combine one-factor-at-a-timeexperiments to a several-factors-at-a-time experiment design which reduces computationaleffort.In Paper C, a surrogate model tailored for RBDO is introduced. It is motivated by appliedsolid mechanics considerations and the use of the first order reliability method to evaluate theprobabilistic constraint. An optimal CAP is furthermore deduced from the surrogate model.In Paper D, the paradigm to use sets of experiments rather than one experiment at atime is challenged. A new procedure called experiments on demand (EoD) is presented. TheEoD procedure utilizes the core of RBDO to quantify the demand for new experiments andaugments it by a D-optimality criterion for added robustness and numerical stability.

FE-Mesh Effect of the Volume Based Weakest-Link Fatigue Probability Applied to a Compressor Blade

When dealing with design process of compressor blades, predominantly deterministic models are used for High Cycle Fatigue (HCF) investigations. The existing scatter in factors such as material inho ...

Numerical prediction of damping in structures with frictional joints

In many vehicle and space structures, friction in bolted joints is the primary source of energy dissipation during vibrations. Several simplified finite element models have proven their capacity to describe the energy dissipation owing to micro-slip in joints. However, these simplified models require extensive physical testing in order to extract model parameters. This paper proposes a methodology based on non-linear FE-analysis of the loading on joints for numerical evaluation of the energy dissipation due to micro-slip. The methodology is extended to allow for computation of modal damping in vibrating structures. Presented applications include studies of an isolated joint as well as a more complex structure subjected to dynamic excitation.

Influence from Contact Pressure Distribution on Energy Dissipation in Bolted Joints

Energy dissipation due to micro-slip in joints is the primary source of damping in many vehicle and space structures. This paper presents results on how the surface topology may be modified to increase the energy dissipation in joints. An analytical solution for general forms of contact pressure of a one-dimensional micro-slip problem is presented. The solution indicates how the contact pressure should be distributed to maximize the energy dissipation. Two dimensional contact pressures are optimized using finite element methods in combination with numerical optimization methods and the results are used to modify the surface topology in bolted joints in order to increase the energy dissipation during loading. The predicted increase of energy dissipation is validated with physical testing. A direct result of the study is a washer with varying thickness increasing the energy dissipation in joints and hence the structural damping of joined structures.

Axisymmetric deformation of transforming ceramic rings

Abstract The present investigation concerns the effect of various possible laws that control transformation of certain multiphase ceramics based on the solution of axisymmetrically deforming thick ring configurations. Stress induced transformation plays a dominant role in toughening, and hence in the strength and fatigue properties of transforming ceramics. The particular transformation law, assumed to hold for multi-axial deformation, affects directly the toughening via the different extent and properties of the process zone developed around macro crack-tips. Phase transformation varies considerably in different microstructures, leading to differences in the macroscopic inelasticity of their stress-strain behavior. It is therefore necessary to develop well controlled multi-axial tests, independently of fracture tests, in order to investigate various propositions for the multi-axial transformation of ceramics. The thick ring configuration under axisymmetric load offers great advantages in experimental, analytical, and numerical assessment of different theories on stress induced transformation. In the present work, some non-linear ring problems were solved exactly accounting for a wide range of transformation laws, and especially for those mostly used in transformation toughening analysis. The exact formulae produced here show that the stress and strain fields in the ring differ considerably with each model. The validity of the various assumptions used in the present analysis stems from well documented micromechanisms reported in experimental papers. The analytic character of the solutions, if combined with experiments, may enable also quantitative predictions of the essential material parameters involved in the modelling of transforming ceramics.