Giuseppe Marulo
Fatigue life assessment of
welded joints
PhD Thesis
University of Pisa
October 2017
University of Pisa
Department of Civil and Industrial Engineering
PhD inIndustrial Engineering
Curriculum vehicle engineering
Fatigue life assessment of welded joints
Thesis by
Giuseppe Marulo
Advisors
Prof. Ing. F. Frendo
Prof. Ing. L. Bertini
Prof. Ing. A. Fatemi
Contents
1 Introduction 1
1.1 History . . . 2
1.2 Motivation . . . 4
1.3 Key topics . . . 5
1.4 Outline of the work . . . 6
I
Literature review
8
2 Basic concepts 9 2.1 Material behaviour under cyclic loads . . . 112.1.1 Cyclic stress-strain relationship . . . 11
2.1.2 Cyclic plasticity models . . . 14
2.1.3 Brittle vs ductile behaviour in fatigue . . . 15
2.2 Basic concepts in fatigue life assessment . . . 17
2.2.1 Crack nucleation and growth . . . 17
2.2.2 Size effect . . . 19
2.2.3 Gradient effect . . . 20
2.2.4 Multiaxial loads . . . 21
2.2.5 Non-proportional loads . . . 22
2.2.6 Variable amplitude loads . . . 24
2.3 Estimation of local elastic-plastic stress-strain at a notch . . . 25
2.3.1 Empirical approximations . . . 25
2.3.2 Analytical formulations . . . 28
2.3.3 Numerical approach . . . 31
2.4 Fatigue life assessment procedure . . . 34
2.4.1 Input parameters identification . . . 34
2.4.2 Stress-strain state evaluation . . . 36
2.4.3 Damage parameter estimation . . . 36
2.4.4 Cycle counting . . . 37
2.4.5 Damage cumulation law . . . 37
2.4.6 Fatigue life assessment . . . 38
3 Fatigue damage parameters for welded joints 39 3.1 Nominal stress method . . . 40
3.2 Local stress-based methods: The notch stress approach (NSA) . . . 43
CONTENTS ii
3.4 Critical plane approaches . . . 50
3.4.1 Fatemi-Socie . . . 51
3.4.2 Carpinteri-Spagnoli . . . 56
3.5 Fracture mechanics based approaches (NSIFs & PSM) . . . 59
3.5.1 Averaged strain energy density: SED . . . 60
3.5.2 Peak stress method . . . 60
II
Reserch activity
64
4 Local stress methods: effects of the actual geometry over the fa-tigue endurance of a welded joint 65 4.1 Evaluation of the fatigue strength of welded joints by the nominal stress, the fictitious notch rounding radius and the peak stress method 66 4.2 Experimental set-up . . . 684.3 Cracks initiation regions and crack paths . . . 70
4.4 Results in terms of nominal stresses . . . 72
4.5 Finite element modelling of the test . . . 73
4.5.1 Weld seam model . . . 74
4.5.2 Global model of the test and its experimental validation . . 75
4.6 Durability analysis based on the notch stress concept and the peak stress method . . . 78
4.7 Stress analysis based on the actual seam weld geometry . . . 82
4.8 Conclusions . . . 85
5 Notch stress approach applied to the fatigue life assessment of thin-walled welded joints made of mild and high strength steels 87 5.1 Definition of effective stresses based on the notch stress approach . 89 5.2 Experimental data set . . . 90
5.3 Finite element models of the specimens and calculation of the effec-tive stresses . . . 93
5.4 Evaluation of the different assessment approaches using effective stresses . . . 95
5.5 Influence of the base material properties . . . 99
5.6 Discussion and conclusions . . . 101
6 Damage cumulation law, life assessment of tests performed under variable amplitude loads 105 6.1 Introduction . . . 105
6.2 Nominal stress method . . . 107
6.3 Experimental set-up . . . 107
6.4 Experimental Tests . . . 109
6.5 Experimental Results . . . 110
CONTENTS iii 7 Comparison of several local stresses methods for the fatigue life
assessment of welded joints 113
7.1 Introduction . . . 113
7.2 Experimental data collection . . . 115
7.2.1 Failure location . . . 118
7.3 Finite element models and analysis . . . 119
7.3.1 NSA models . . . 122
7.3.2 PSM models and sub-models . . . 123
7.4 Results and discussion . . . 125
7.5 Conclusions . . . 127
8 The Fatemi-Socie critical plane method: computational aspects for the application to welded joints 129 8.1 Specimen description and experimental set-up . . . 131
8.2 Experimental tests . . . 132
8.3 Fatigue life assessment . . . 133
8.3.1 Finite element simulations . . . 133
8.3.2 Elasto-plastic stress-strain estimation . . . 135
8.3.3 Fatemi-socie damage parameter evaluation . . . 136
8.4 Discussion and conclusions . . . 137
8.4.1 Mean stress in bending and torsion . . . 138
8.4.2 Phase angle and nominal to shear stress ratio effects on com-bined load tests . . . 139
8.4.3 Conclusions . . . 140
III
Conclusions
141
9 Conclusions and future developments 142 9.1 Conclusions . . . 1429.2 Ongoing research activity and future developments . . . 143
9.2.1 Fatemi-Socie critical plane approach applied to variable am-plitude loading . . . 144
9.2.2 Residual stresses in fillet welded pipe-to-plate joints . . . 145
9.2.3 Residual stresses evaluation . . . 146
Chapter 1
Introduction
Fatigue is one of the most common failure mechanisms of structural components. Many fatigue failures have become popular throughout engineering history since the tragic railroad accidents that motivated Wöhler to begin the first investigations of the fatigue damage process. Since then a huge deal of work has been carried out ot the subject by several generations of engineers leading to a deeper comprehension of the complex phenomena related to the damages occurring in structural components due to time varying loads. However, many aspects of the subject remains unclear and several questions are still open for discussion.
The present work tries to give an answer to some of those questions. In particular, it is focused on the fatigue life assessment under multiaxial and variable amplitude loads with regard to the application to welded joint connections.
Fatigue is a classical topic in engineering design and a subject of wide practical interest since it is one of the most common causes of failure for structural components as discussed in section 1.2.
According to ASTM [1] fatigue can be defined as:
The process of progressive localized permanent structural change occurring in a material subjected to conditions that produce fluctuating stresses and strains at some point or points and that may culminate in cracks or complete fracture after a sufficient number of fluctuations.
The ASTM definition is truthfully analysed by Stephens et al. in their book about metal fatigue [2], where six keywords in the ASTM definition are individuated, each one of which describes a key aspect related to the fatigue damage process:
• progressive, this word refers to the fact that fatigue does require a certain time interval in order to produce a damage in the material;
• localized, this term comes from the observation that fatigue damage takes place at a specific location of the component and involves only a local volume
of material. The local feature of fatigue implies that even a small defect can produce a marked effect on the fatigue endurance of a large structure; • permanent, this implies that fatigue is an irreversible phenomenon.
• fluctuating, refers to the time varying nature of a fatigue damaging load. This is a necessary condition since no fatigue damage can occur in a static load condition whatever the load level is;
• cracks, are the ultimate cause of all fatigue failures. Under repeated loading a fatigue crack grows till the remaining undamaged material is not able to withstand the applied load;
• fracture, is the last stage of fatigue damage, where the component is separated into two or more parts by the applied load.
1.1
History
The industrial revolution in the beginning of the nineteenth century leaded to the production of metallic rotating machinery powered by steam engines. Those components were the first to experience a number of loading cycles capable of producing fatigue failures. In particular, the sudden fatigue failure of railroad axles caused several catastrophic accidents and motivated engineers to understand the newly discovered phenomenon in order to prevent similar disasters to happen again. Thus, fatigue life assessment is a subject that has concerned engineers for almost two centuries, it appears evident that any development of the field has to be based on a sound knowledge of the remarkable milestones that have characterized its development. That is provided here as a brief summary of the principal achievements in the history of fatigue life assessment. More exhaustive historical overviews can be found in many textbooks such as [2].
The entries reported here are listed in chronological order, for each one the date and the author’s name are indicated besides a concise description of the achievement itself. Where possible references to the original literature works are provided.
1840 Wöhler : after several railroad accidents, made the first studies on railroad axles failures. He introduced the use of S–N curves and postulated the fatigue limit concept [3];
1870s Goodman and Gerber : studied the effects of mean stresses on fatigue proposing the well known diagrams that still today are named after them. 1886 Baushinger in 1886: made some early considerations on cyclic plasticity
discovering that the yield strength results to be modified by inelastic deformations [4].
1900s Ewing and Humfrey: described the slip bands mechanism leading to the formation of microcracks in ductile materials [5].
1910 Basquin: proposed a power law relationship of the S-N curve in the finite fatigue life region, which results in a straight line when fatigue data are plotted in a log-log graph [6].
1921 Griffith: published his study on brittle fracture [7] developing the basis for fracture mechanics.
1924 Palmgren: proposed the linear damage rule for cumulative damage, it’s use was later made popular by Miner in 1945 [8, 86]
1930s Almen: demonstrated the beneficial effect of compressive residual stresses on fatigue.
1935 Gough and Pollard : proposed their ellipse criteria for multiaxial fatigue assessment [9].
1937 Neuber : described the gradient effect on fatigue damage mechanism and formulated an analytical formulation for the stress evaluation over blunt notches [10].
1939 Gassner : introduced variable amplitude testing [11].
1953 Peterson: First edition of the book on stress concentration factors Kt and relation with Kf [12].
1958 Irwin: proposed the use of the stress intensity factor KI and coined the term "fracture mechanics" [13].
1960s Manson-Coffin: dealt with low cycle fatigue proposing a relationship between plastic deformation and fatigue life [14].
1961 Paris: formulated the power law relation between stress intensity (∆K) factor and crack growth rate (dNda) known today as Paris law [15].
1968 Matsuishi-Endo: proposed the rain flow cycle counting for variable amplitude loads [16].
1970 Elber : described a physical model for crack closure mechanism [17]. 1973 Brown-Miller : developed one of the first critical plane approaches [18].
1.2
Motivation
The practical interest in fatigue is strictly related to the vast number of failures caused this phenomenon and their high costs for the society. The history of fatigue and fracture mechanics studies is strictly related to catastrophic accidents that demanded high costs not only in terms of economic losses but also in terms of human lives. Since from the fists studies on fatigue damage conducted by Whoeler, that started it’s work in order to understand the sudden failure originated in train axles that caused several railroad incidents.
The development of fracture mechanics was motivated by another series of tragic losses: the sudden fracture of several liberty ships that occurred during World War II. This historical fact is particularly relevant to the current work since those were the first ships whose hull was manufactured by welding, an innovative joining technique for that times. In that period US shipyards were requested to produce a large number of ships as fast and cheap as possible. For that reason welding was substituted to riveting as the main joining technique for the ship’s hull plates. Welding allowed for a faster production and reduced the required amount of material since no overlapping was necessary between the to plates to be joined. Unfortunately, several catastrophic failure were registered and a number of those ships literally split in two halves due to brittle fracture caused by the local stress concentrations in the welded joints and by the low fracture toughness of the material in the cold environment of the northern Atlantic sea.
Another failure, that of the Comet, was the root for the decision to shift from fail-safe to safe-life strategies in the design of aircraft components. The Comet was the first jet propelled aircraft to take service for passenger transportation. Two Comets failed due to cracks emanating from sharp corners on a window opening in the fuselage. Even if the plane had been thoughtfully fatigue tested prior to service, the conducted test program resulted to be inappropriate and produced an overconfident fatigue life assessment.
A lot of improvements have been achieved in life assessment procedures, however, fatigue is still among the main failure causes for many components of practical interest. As an example, in the automotive sector 50-90% of the total failures is attributed fo fatigue induced damage [2]. A study conducted in 1978 about the costs related to fracture in the United States [19] indicated the astonishing loss of $119 billion due to this failure mechanism, which was circa equal to 4% of US gross national product at the time. Those numbers witnesses that fatigue is still an actual topic and further improvements are needed in order to reduce it’s costs for the society.
1.3
Key topics
The current work aims to investigate the fatigue life assessment of welded joints under the effects of multiaxial loads. Therefore, it is focused on two main research topics, which are multiaxial fatigue and it’s effects on the endurable life of welded joints. Both topics are characterized by a wide practical interest since welding is an extremely diffused joining technique and multiaxial stress-strain conditions are encountered in many applications.
Welding is one of the most spread joining techniques in industrial applications since it offers a number of practical advantages. Among the others, it allows complex geometries to be obtained, the production process can be completely automated and it allows to a weight reduction, if compared to other techniques such as riveting or fastening. Many structures are mainly produced trough welding, as an example the parts constituting an automotive or motorcycle chassis are joined by welding. This joining technique is found in very large structures such as ship’s hulls and civil structures like steel framed bridges and skyscrapers. But, welding is also used in the manufacturing of very small components, like: sensors, solenoids, switches, thermocouples, batteries, metal bellows etc.
Whatever the size and the application, welded joints shows some common characteristics. First of all, the creation of a welded connection does always require a certain amount of energy to be involved which is generally transferred to the joint for an heat source. Thus, during the welding process, the joint temperature arises, in most cases, till a partial fusion of the base material is obtained. The thermal cycle associated with the heat generation in the manufacturing process is responsible for alterations in the mechanical properties of the materials constituting the joint and for a large amount of residual stresses that are generally registered in welded seams. Plus, the high energetic nature of the process is responsible for a large variance in several features of the finite product like geometry and mechanical properties. This intrinsic variance is reflected in the characteristic wide scatter of the fatigue endurance data that can be obtained through experimental investigations. Finally, welded joint shows the presence of severe notches in the weld seam cross section and, often, crack like defects are encountered in the weld region. Such geometrical features give rise to local stress concentrations and are a preferential location for crack nucleation and growth. Therefore, a notched welded seam can have a relevant effect on the fatigue life of the structure in which it is present.
The multiaxiality of the stress-strain state results to be quite linked to the presence of notches. On a plain component, the application of at least two loads acting on different directions is required in order to obtain a multiaxial load. Even if the presence on more loads is not uncommon in practical applications, it is not encountered in the majority of components. On a notched component, instead, the geometrical stress concentration can induce a multiaxial stress state even under an uniaxial loading action depending on the notch geometry and loading direction. This conclusion coupled with the observation that notches are critical areas for crack nucleation makes the study of multiaxial fatigue a topic of wide practical interest.
1.4
Outline of the work
Proper design against fatigue failure should be made by both analysis and testing [2] since each of those steps alone is insufficient in achieving reliable results. Testing is a fundamental tool to be used for product durability determination, but it is not suited for the development of new components. On the other hand, the analysis that can be done by current fatigue damage models is reliable enough in order to assess the fatigue endurance of safety critical parts. The term model itself suggests that the physical reality is simplified in its mathematical description which is governed by a finite number of equations, even when dedicated computer aided design is adopted the number of degrees of freedom considered can be increased but still remains finite. In the real world, instead, the fatigue damage mechanism involves a number of complex interactions between different physical phenomena that can hardly be condensed into a descriptive model. Even in simple everyday applications several synergetic aspects are involved in fatigue, like variable amplitude loading, temperature dependence and stress-corrosion cracking. Also, when welded joints are investigated their intrinsic variance can hardly be included in any predictive model.
The current work is divided into three main parts. The first part, will be dedicated to a literature review. An analysis of the basic concepts related to fatigue damage is presented in chapter 2 while a review on the current state of the art in terms of fatigue damage parameters evaluation is exposed in chapter 3. Both of those introductory chapters are focused on the application to welded joints.
The research activity that will be discussed in the second part of this work is composed of both testing and analysis. In particular, five aspects related to fatigue life evaluation will be discussed in as many chapters (chap. 4,5,6,7 and 8). Each chapter is dedicated to a specific topic regarding the fatigue life assessment of welded joints, that is discussed by modelling and analysis performed according to regulation codes and other techniques derived from the literature. The analysis results are compared with experimental evidences in order to evaluate their reliability. The experimental data presented in the following chapters were mainly provided by the testing activity conducted at the laboratories of the University of Pisa.
In particular, the first three chapters (chap. 4,5 and 6) are targeted to evaluate peculiar aspects of the fatigue life assessment when welded joints are involved. While the following two chapters are dedicated to the fatigue life assessment by locally defined damage parameters.
Chapter 4 points out the relevant improvements that can be obtained going from global to local models. Also, the actual geometry of a common fillet weld seam was thoughtfully investigated from a fatigue point of view.
In chapter 5, the fatigue life of welded joints made of different materials is evaluated. The selected materials were all steels which use in the fabrication of automotive chassis frames is widely spread. Due to the peculiar application, only joints with a thin plate thickness (below 2 mm) were included in this investigation. A local stress based damage parameter is adopted for the evaluations of this chapter: the notch stress approach. Its application was combined with several stress averaging techniques.
In chapter 6, the effects of variable amplitude loads on the fatigue life of welded joints is discussed on the basis a testing champaign on pipe-to-plate welded joints under block loading. The analysis presented here is performed according to global nominal stresses. However, a more in deep analysis based on the Fatemi-Socie critical plane approach is currently under development and is briefly presented in the future developments section of chapter 9.
Chapter 7 presents a comparison of two local fatigue life assessment methods, the notch stress approach and the peak stress method. The performance of those two analysis models is evaluated thanks to a vast database of experimental results coming from different specimen geometries and constant amplitude load configurations.
In chapter 8 the application of the Fatemi-Socie critical plane approach to the fatigue life assessment of welded joints is discussed. An evaluation routine is developed in order to comply with the specific characteristics of the joints under investigation.
Most of the work presented here has been published on international journals [20, 21] or in the procedia of international conferences [22], while some other sections are still under review for journal publication (chap. 7) or have been accepted as conference papers (chap. 8).
Finally, the third and last part of the current work contains some conclusive considerations and a brief outlook on future developments, some of which are currently ongoing research.
Part I
Chapter 2
Basic concepts
This first chapter is dedicated to point out some peculiar aspect related to the fatigue damage phenomenon and the assessment procedures that are nowadays renowned as standard practices. The chapter is divided into four sections, each addressing a specific topic that is thought to be relevant in the development of the present work:
• material behaviour,
• local stress-strain estimation over a notch,
• basic concepts related to fatigue damage process, • general fatigue life assessment procedure.
The first point to be treated will be the the material behaviour under cyclic loads. It is common practice to characterize a metallic material by standardized static tests such as the tensile test and fracture toughness tests due to the relative simplicity of those tests. However, it is well known that the material response to cyclic loads can be substantially different compared to static ones. In particular, cyclic strain hardening (or softening) is often developed by metals, this modification in the stress-strain response can have a relevant effect on the endurable fatigue actions. Furthermore, the brittle vs ductile behaviour from a fatigue point of view will be analysed. It will be shown that not only the material properties influence the transition between those two fields but also the load conditions and the components geometry plays a role in defining the material behaviour.
The second section, will be dedicated to the discussion of some basic concepts related to the fatigue damage process since a deep understanding of the physical phenomena occurring in the cracking mechanisms is a key aspect for a reliable life assessment. This section will point out some assumptions that are at the basis of all the work presented in the following chapters, such as the hypothesis that crack propagation life is small compared to crack nucleation one. This is a standard practice in welded joints life evaluation, however, its validity can not be taken for granted in any kind of application and should be always verified.
In the third section, the estimation of local stress-strains on a notched component will be discussed. This aspect is particularly pertinent to the current work since sharp notches are always encountered on a welded seam. The fatigue
endurance of such components is strictly related to the stress concentration occurring at the notch tip, not only in terms of maximum stress-strain value but also in their distribution.
Finally, a flow chart describing the complete fatigue life assessment procedure is presented in the last section of this chapter. The current work will be mainly focused on one of the blocks in this flow chart: the damage parameter evaluation. However, it has to be taken in mind that the ultimate goal is a good endurable life evaluation that can be obtained only if a good balance between all the steps is achieved. As an example, in analysing the life of variable amplitude loaded components the damage parameter, cycle counting algorithm and cumulation law should always be chosen to be in good accordance with each other.
2.1
Material behaviour under cyclic loads
The current section is developed with particular regard to steels since all the actual components analysed in the following chapters are made from this material, however most of the presented considerations can be extended to generic metallic materials like aluminium or titanium alloys.
Fatigue by its nature is always related to time varying loads, therefore, one of the first aspect to be considered when approaching fatigue damage evaluation is the response of materials to cyclic loading. This proves to be a key aspect in fatigue analysis since the relationship between stresses and strains can be substantially different going from static to time varying loads. Many metals, in facts, show a relevant degree of strain hardening (or softening) under cyclic loads. This implies that the stress response to an applied strain can be significantly greater (or lower) compared to what could be expected from static load condition. The stress-strain response has a relevant effect on fatigue life, any damage parameter can lead to reliable results only if the material behaviour is correctly taken into account.
Also, a brief excursus on plasticity models for fatigue applications is presented. This would be an extremely wide and rather complex subject that would demand a much more extended discussion but this would be outside the scope of the current work. Only some basic information will be given in the following section, however references to more detailed works will be given since this is a fundamental step for a robust fatigue life assessment.
Finally, the ductile versus brittle behaviour will be reviewed from a fatigue point of view. It will be shown how not only the material properties are responsible for the two different cracking natures but also the applied load and components geometry plays a significant role to this regard. Ductile and brittle behaviours are related to different fatigue damage mechanisms. A ductile failure, generally, shows a relevant degree of plastic deformation and a transgranular microcracking preceded by the formation of permanent slip bands. This phenomenon is governed by the amount of tangential strain imposed to the crystal structure of the grain. On the other hand, brittle failures commonly occurs without any relevant plastic deformation and intergranular cracks are generally observed. In such conditions the main fatigue action is assumed to be the normal stress acting perpendicularly to the advancing crack.
2.1.1
Cyclic stress-strain relationship
Comparing cyclic stress-strain relationships to static ones two phenomena are to be discussed:
• cyclic hardening,
• non-proportional hardening.
An example of cyclic strain-hardening is presented in fig. 2.1 for a S355JR steel. In the graph, both the stress-strain relationship obtained from a conventional tensile test (red curve) and the experimentally derived cyclic stress-strain law (blue
curve) are plotted. In fig. 2.1, some experimentally measured stabilized hysteresis loops for different applied strain ranges (cyan curves) are also reported.
It can be seen as the tested material shows an initial softening, for low applied strains. This is followed by a marked hardening when the applied strain is increased. For this particular steel, the stress measured under cyclic loads at an applied strain equal to = 0.02 resulted increased of the 25% compared to the static stress measured at the same strain value. This constitutes a relevant difference in material behaviour that, if neglected, can significantly deteriorate the stress-strain evaluation and therefore the fatigue life assessment. It has to be noted, that a 2% elongation for the selected steel is a relatively modest value since it presents an elongation at break equal to A = 18%, even if it involves a relevant amount of inelastic deformation.
0.03 0.02 0.01 0.00 0.01 0.02 0.03
True strain
(²) 600 400 200 0 200 400 600 800Tr
ue
st
re
ss
( σ )Material: S355JR
Cyclic stress-strain relationship
Static tensile test
Figure 2.1: Tensile test results and experimentally derived cyclic σ − relation for S355JR steel.
The experimental determination of the cyclic-stress strain relation can be conducted according to standards E606 from ASME and J1099 form SAE [2]. There exists three test procedures, namely: the companion test method, the incremental test method and multiple step test method. Those three methods differ in the number of required specimens and in the testing procedure. Several specimens are required for the companion test method, each one of those is tested at a fixed strain amplitude. The cyclic loading is carried out until failure occurs. In this case the hysteresis loops are reordered at half the fatigue life of the specimen. The incremental and multiple step test methods, instead, allows to determine the cyclic stress-strain curve testing a single specimen. The multiple step is similar to the companion method since a fixed amplitude strain is applied to the specimen, but in this case the test is carried out till a stabilized hysteresis loop is measured. Then the applied strain range is increased and the procedure repeated. The incremental step method is conducted applying repeated blocks of incrementally increasing and decreasing strains. The hysteresis loop are registered after stabilization, that generally occurs after several load blocks are applied. In any case, the cyclic stress-strain relationship is obtained as the line
that connects the vertex of the measured stabilized hysteresis loops. This step is often performed by a least square regression analysis.
Further hardening of the material can be observed when the applied loading path involves a rotation of the principal strain axes during cyclic loading, such conditions are referred to as proportional loading [23]. A special case of non-proportional load is constituted by out-of-phase loads. This term refers to cyclic loading histories obtained by the simultaneous application of two independent loads with sinusoidal waveforms and a phase shift between the two signals.
The maximum degree of non proportional hardening is generally registered for 90◦ out-of-phase loading [23], which is also assumed as the most damaging (from a fatigue prospective) load condition by many authors and then selected for fatigue testing [24–27]
The non-proportional hardening (and therefore its magnitude) is related to the microstructure of each specific material. A greater number of slip planes with different orientations is activated by the rotation of the principal strain axes compared to proportional loads. The degree of non-proportional hardening can be quantified by the factor α given by the ratio between the equivalent stresses measured in 90◦ out-of-phase versus proportional loading at high applied strains (flat part of the cyclic stress strain relationship) [23]. The amount of non-proportional hardening for less severe load conditions (e.g. 45◦ out-of-phase) can be evaluated by interpolation based on the non-proportionality factor (defined in sec. 2.2.5) of the selected load.
Ramberg-Osgood relation
A mathematical representation of stress-strain relationship described in the previous section is often adopted in order to allow a description of the curve by a limited number of material dependent constants. One of the most spread mathematical representations is the description proposed by Ramberg and Osgood [28], which is given by equation 2.1 and it is graphically represented in figure 2.2. ∆ 2 = ∆σ 2E + ∆σ 2K0 1/n0 (2.1) Thanks to Ramberg-Osgood (RO) formulation the stress-strain material behaviour can be described by three parameters namely: the Young’s modulus E, the strength coefficient K0 and the strain hardening exponent n0. The wide spread use of the RO is related to the fact that it gives a continuous and regular curve with no discontinuities in the transition between elastic and plastic regimens. This constitutes a useful feature in the development of analytical procedures, such as the Neuber’s rule (sec. 2.3.1).
The total strain is represented by the RO as the summation of two terms (eq 2.1). A non linear term, on the right hand side ( 2K∆σ0
1/n0
), is added to the linear elastic equation on the left hand side (∆σ2E). This leads to a certain degree of approximation in the linear elastic region where a non-linear term is added to the linear formulation. The error coming from this approximation is made relatively
True strain
(²)Tr
ue
st
re
ss
( σ )Ramberg-Osgood equation plot
Figure 2.2: Ramberg-Osgood relation.
small by the exponential formulation on the non-linear term that assumes limited values at low strains.
2.1.2
Cyclic plasticity models
The inelastic behaviour of the material is a relevant aspect involved in the fatigue damage process, especially when notched components made of ductile behaving materials are taken into account. The importance of cyclic plasticity on the multiaxial life assessment is remarked by the following statement from Socie and Marquis [23].
. . .understanding multiaxial cyclic deformation is the key to unlocking the secrets of multiaxial fatigue.
However a comprehensive review of the literature regarding cyclic plasticity models would be an extensive work due to the complexity of the subject and the vast amount of technical literature on it. Furthermore, it would be outside the scopes of the present work as the topic is discussed in several text books [23, 29]. However, a brief review is presented here.
When the incremental theory of plasticity is applied, for each load variation it has to be evaluated if plastic deformation is going to happen and, eventually, it’s amount. Finally, when plastic deformation occurs, the variation of the yield function has to be evaluated. Therefore, any plasticity model is composed of the three conceptual blocks:
• a yield function; • a flow rule; • a hardening rule.
The yield function describes the combination of stress that will lead to plastic deformation, thus indicates if the applied load under examination is going to
produce plastic deformations. The most used yield function is the renowned deviatoric stress proposed by von Mises.
The flow rule is the relationship between stress and plastic strains increments, this tool allows to evaluate the amount of plastic deformation that is developed. Finally, the hardening rule describes how the yield criterion changes with plastic straining. Among the possible hardening rules, the two simpler ones are the so called isotropic and kinematic one.
Isotropic hardening assumes that the shape and location of the yield function does not change with plastic straining, but only it’s dimensions increase. The kinematic term, instead, denotes hardening rules where the yield function is rigidly shifted but keeps its shape and dimensions. Many structural materials show a behaviour which is composed of both kinematic and isotropic hardening.
The hardening rule definition for a specific material is, also, influenced by the applied load conditions. For general application an isotropic hardening rule is a good assumption when stabilized behaviour due to constant amplitude loads is accounted for. In variable amplitude load scenarios, instead, this could lead to misinterpretation of the actual strain/stress history. [23]
A more general description of cyclic plasticity can be achieved by the use of the so called multiple surface models, among which we can find the one proposed by Mroz [30, 31] and later developed by [32]. In the multiple surface model both isotropic and kinematic hardening are accounted for. This kind of model shows the advantage to correctly predict the Baushinger effect for Masing behaving materials [33].
2.1.3
Brittle vs ductile behaviour in fatigue
It is common practice to regard brittle or ductile behaviours only as material dependant characteristics. However this is not entirely correct when fatigue damage is taken into account since components made of the same material can fail either in a ductile or in a brittle way according to the load conditions and the components geometry. High load levels, usually, tend to produce brittle failures while ductile mechanisms are more likely to take place under moderate loads. Also, brittle fracture is more likely to happen in severely notched components due to the stress concentration that is present in such cases.
An useful index to discern between brittle or ductile behaviour in fatigue, based on the material characteristics, is the ratio given by the endurance fatigue limit in fully reversed torsion (τaf) versus fully reversed bending (σaf) [34]. Usually, a brittle behaviour is observed for materials that present a fatigue limit ratio in the range √1
3 ≤
τaf
σaf ≤ 1. For ductile behaving materials, instead, the
endurable stresses in bending and torsion tend to be closer and their ratio is generally smaller τaf
σaf ≤
1 √
3. However, a certain proof of ductile or brittle behaviour can only be obtained from the analysis of the crack nucleation zone. In ductile conditions, in facts, the damage mechanism is dominated by shear straining. Local plastic deformations are produced by dislocation movements on planes oriented in the maximum shear strain direction, generally on the surface of the component. Under cyclic loading, said dislocations produce a damage in the
crystal grain structure until permanent slip bands [35] are created. If cyclic loading is continued, those slip band constitute a preferential location for crack nucleation. Therefore, cracks are observed to nucleate on maximum shear strain planes. As an example, a round bar under cyclic axial load will show cracks oriented at 45◦ with respect to the specimen axis if a ductile damage mechanism takes place. Although kinematically irreversible microscopic deformations are still possible due to several mechanims such as microcracking, interfacial sliding etc. [35] In hard materials, crack nucleation is, generally, associated with bond rupture that takes place with little or no plastic deformation. This phenomenon is governed by normal stress and the cracks tend to nucleate on the planes that undergo the highest variations in terms of normal stress.
The two described mechanism are influenced by different fatigue actions, the tangential strain variation is the key factor for ductile crack nucleation while brittle solids cracking is more dependant on normal stresses. This aspect is also reflected by fatigue damage parameters; as an example the Fatemi-Socie critical plane approach [36] for ductile materials is based on cyclic plastic tangential strain. The critical plane model for hard materials proposed by Carpinteri-Spagnoli [34], instead, assumes that the normal stress is the main damaging action.
Another aspect that can influence the damage mechanism is the presence of a flaw or a notch. It is well known, in facts, that a brittle crack propagation can take place even for ductile materials depending on the existing crack geometry. Linear elastic fracture mechanics is often employed to correlate crack geometry, material fracture toughness and crack propagation rate in cases where the the amount of plastic deformation is negligible due to the high stress gradient related to the concentration factor on the tip of a sharp crack.
2.2
Basic concepts in fatigue life assessment
For a reliable and robust fatigue life assessment it is mandatory to understand some basic principle related to the damage process itself. For this reason the main aspects related to the fatigue damage mechanism are reported in the following sections. The attention will be focused on five topics, each one related to a specific aspect inherent the fatigue phenomenon:
• crack nucleation and growth, • size effect,
• gradient effect, • multiaxial loads,
• non-proportional loads, • variable amplitude loads.
2.2.1
Crack nucleation and growth
The damage mechanism is a complex phenomena that is widely described in literature. In this section only some aspects of it will be exposed, the interested reader is referred to some classic text books for more detailed informations [23, 35, 37].
One of the main aspects related to the damage mechanism is that it can be divided into several phases. Different models have been proposed in the literature, with a number of phases ranging from three to five [38–40]. In the current work, it will be described the simplest model composed of three phases [41]:
• crack nucleation; • crack propagation; • final failure.
In the first nucleation phase one or more initial cracks are produced (i.e. nucleated) in the material volume, this phase is generally termed as "stage I". There are several nucleation mechanisms possible (sec.2.1.3). A ductile behaving material generally shows the formation of permanent slip bands in grains along the surface of the component. In brittle materials, instead, the nucleation can take place on grain boundaries with no plastic deformation. The process that leads to the formation of permanent slip bands in ductile materials was firstly observed by Ewing and Humfrey in 1903 [5]. Plastic deformation occurs in grains that show crystallographic planes with favourable orientations. During cyclic loading the slip bands grow in size and number, until they eventually coalesce into a single fatigue crack.
The nucleation phase is generally assumed to be ended when a fatigue crack reaches the length of 1 mm. This numerical value is commonly adopted since it
is comparable to the minimum crack size that can be detected by standard non destructive techniques.
Other authors (e.g. [38]) have proposed more detailed descriptions where the nucleation phase is taught to end at smaller crack lengths and additional phases of microstructurally or physically short cracks propagation are taken into account. The damage process then proceeds with the long crack propagation phase, usually referred to as "stage II". In this phase the crack grows until the cross section of the component is to small to withstand the applied external load at its maximum value. The crack length in final failure condition is generally termed critical ac. The crack growth process is commonly analysed with fracture mechanics considerations.
Lastly, the final failure, can be obtained through several mechanisms depending on the component and its applications. It is common to observe brittle fracture of the residual cross section, but also generalised yielding of the un-cracked ligament can produce excessive deformation and then failure of the component.
The relative length of the nucleation and propagation phases can be relevantly different depending on the component geometry, the material behaviour and the applied load history. An example of fatigue lives with different repartitions between nucleation and propagation is shown in fig. 2.3.
Life fraction
0 20 40 60 80 100
crack nucleation phase crack propagation phase
Figure 2.3: Life fraction examples.
There exists a relation between the applied load level and the relative fraction of life spent over the two phases. When a load great enough to produce plastic deformation is applied to the component,the nucleation For lower loads the life fraction of the nucleation phase tends to increase. This explains why nucleation based damage models are generally preferred in high cycle fatigue.
Also, the material brittle or ductile behaviour can affect the relative phase lengths. As an example, in a very brittle material the final failure can occur for cleavage fracture even when a relatively small crack is present. On the other side, a ductile behaving material can withstand the presence of cracks with a relative high length.
Finally, the effects of component’s dimensions over the life distribution is discussed in the following section (sec. 2.2.2).
A wide number of fatigue life assessment procedures is designed in order to account only for the crack nucleation phase life fraction, under the common hypothesis that the propagation phase is carried out in a small number of cycles and thus can be neglected. This is called safe life design. Other criteria, instead, focus on the crack propagation life fraction alone, disregarding the nucleation
phase. Such an assumption is justified when crack like defects can be found in the component due to the manufacture process or previous load application. This is, generally, the application field of fracture mechanics and damage tolerant design. In this case, one ore more cracks are thought to be already present in component before it starts its service life. The initial crack size evaluation is a relevant topic in this approach since the actual crack size greatly influences the crack growth ratio and the residual fatigue life. When non destructive testing is performed prior to installation or at fixed time intervals, the initial crack size is supposed to be equal to the smallest defect that can be revealed by the testing.
However, the validity of said hypotheses should never be taken for granted and checked for each application since their validity can be influenced by many factors. Generally speaking, a clear distinction between nucleation and propagation phase is not available. Usually, the distinction is made based upon crack length. Also, it is common practice to assume a crack length of a = 1 mm as discriminating value, however this value is not related to any damage mechanism but only to the minimum detectable defect size offered by standard non destructive testing (NDT) devices.
A transition crack length related to the damage mechanism is the well-known a0 proposed by El Haddad ([42]). It is defined as the crack length at which the fatigue limit stress becomes greater than the endurable gross stress for crack propagation according to linear elastic fracture mechanics. The graphical representation of the transition crack length on the Kitagawa-Takahashi diagram [43] is represented in many textbooks.
2.2.2
Size effect
Even if the fatigue damage process is an localized phenomenon the component’s dimensions show a relevant influence on the endurable fatigue life.
It is wide known that bigger components tends to show a shorter fatigue life compared to smaller ones loaded at the same stress levels. This can mostly be explained based on probability considerations. In facts, a bigger component is more likely to show a geometrical or metallurgical defect able to enhance the fatigue damage process thus reducing its expected life. The same probabilistic considerations does explain the different fatigue lives experience by specimens loaded in rotating or alternating bending at the same stress ranges [44]. The fatigue actions in the two cases are exactly the same except than the volume of material that undergoes the maximum stress range is different. Taking as an example a round notched bar specimen, any point on the notch will experience the maximum stress variation over a cycle disregarding its angular position if rotating bending is applied. Instead, when an alternating bending load is considered only two points (on the bending plane) undergoes the maximum stress range. Thus, in rotating bending it is more likely to solicit a pre-existing defect over the notch itself.
Furthermore, the component’s size has and effect on both the nucleation and propagation phases of the damage mechanism. The nucleation phase is an extremely localized phenomenon, it can take place at a single grain or at a group of adjacent grains involving a limited volume of material (less that 1 mm3).
Therefore, the component dimensions have no effect on the nucleation process itself but rather they affect the probability that a grain with unfavourable orientation or a crystallographic defect is present thus the probability of crack nucleation increases with increasing component’s dimensions. The above mentioned considerations leaded to the formulation of the highly stressed volume [45] hypothesis, where the fatigue life is related to the volume of material that experiences a solicitation greater than a fixed percentage of the maximum value.
The size effect on the propagation phase is more evident, it is related to the amount of crack growth that is endurable by the component before failure occurs. However, even if more evident, it can be less relevant than in crack nucleation depending on the specific component and load condition. First of all the life portion spent in each phase has to be taken into account. Often the crack nucleation phase constitutes a preponderant fraction of the whole damage process. In such cases the crack propagation can be neglected (this is a common hypothesis in fatigue life assessment of many components including welded joints) and so it’s dependence on the component size. On the other hand, even if the crack propagation is a non negligible parto of the fatigue life, the component’s dimensions could have a limited effect on the whole fatigue life. In facts, the crack growth ratio exponentially increases with the actual crack size as the Paris power law states [15, 37]. Therefore, the crack propagation can be quite fast when the crack has already reached a remarkable size and the number of cycles spent in long crack propagation can be relatively short.
Generally speaking, for small components with high stress levels and no initial imperfections the crack nucleation phase tends to be much greater than the crack propagation one. The opposite behaviour is shown by bigger components with lower stress levels which are generally produced with technological processes that can not exclude the presence of initial imperfections. Unfortunately, real cases tend to be somewhere in between the two opposite behaviours.
2.2.3
Gradient effect
The first to address the gradient effect on fatigue life was Neuber in its celebrated work on the stress distributions over blunt notches [10]. He stated that the fatigue endurance where severe stress gradients are present is not related to the maximum elastic stress but rather to an effective stress value that can be obtained from averaging over a characteristic material length. Neuber’s work constitutes the theoretical background for one of the most spread fatigue life assessment methods of welded joints: the Notch Stress Method (sec. 3.2) This concept can be easily understood taking as an for example a sharp notched component where the elastic stress would reach an infinite value even for low applied stresses while the structure retains a finite fatigue life.
The observation that elastic notch stress alone is insufficient for the estimation if fatigue life of structures has been widely adopted in the literature. Local stress based approaches for the fatigue assessment of notched structures, such as welded joints, are often performed converting the elastic-linear stress field in the notch
area into an effective stress. This can be formally stated as:
σe = ησk, (2.2)
where σk is the elastic notch stress, while η is the notch sensitivity factor and σe is the stress value that is thought to be effective from a fatigue point of view. It is theoretically possible to define an infinite number of functions capable of converting the elastic stress field into a fatigue effective one. Looking at the same matter from another prospective, the nominal stress σncan be related to the notch and effective stress respectively:
σk= Ktσn, σe= Kfσn. (2.3) Therefore the notch sensitivity coefficient η from eq. 2.4 can be thought as the ratio between the fatigue notch factor and the notch stress concentration factor:
η = Kf Kt
. (2.4)
The same physical phenomenon can be described by means of the notch sensitivity factor reported by several authors [29, 44]:
q = Kf − 1 Kt− 1
. (2.5)
The notch sensitivity factor is assumed as a material dependent constant and its value is included in the interval 0 ≤ q ≤ 1, where q = 0 implies that there is no effect of the stress concentration on fatigue life (Kf = 1), while the maximum degree of notch sensitivity (Kf = Kt) is obtained for q = 1.
In [46, 47] a general theory based on the functional analysis is developed, in order to relate the elastic stress to its effective value by means of stress averaging. The general form of the stress averaging procedure is given by eq. 2.6, where the vector x denotes the coordinates of a single point in the structure, Ω is the integration domain and W a weight function.
σe = Z
Ω
W σ dx, x ∈ Ω (2.6)
Most of the stress averaging techniques in the literature can be obtained giving opportune formulations to the variables of eq. 2.6. An example of that is provided by the line integral proposed by Neuber [10] and the point method described by Taylor in its theory of critical distances [48]. Those two stress averaging methods [49] have been used in combination with the notch stress approach proposed by [50] for the fatigue life assessment of thin welded joints where a small reference radius is required. This technique is more extensively discussed in chapter 5 and its performance is evaluated by means of a comparison with experimental results.
2.2.4
Multiaxial loads
The term multiaxial load is used to describe a loading condition that produces a stress state where more than one principal stress is different from zero. This term
is used in contrast to uniaxial loading where only one principal stress is present and the material undergoes a loading acting only in a specific direction, a single axis. The vast majority of materials testing is performed under uniaxial loads since they can be applied by means of simpler testing devices. An example of that is given by the tensile test and the mode I fracture toughness testing in plane stress conditions. Therefore, equivalent stress criteria are adopted in order to compare the effects of an applied multiaxial stresses to the experimentally derived uniaxial endurable action.
Often times, the same approach is extended to fatigue life estimation, but unfortunately with poorer results. While, an equivalent stress (e.g. deviatoric stress from von Mises) can successfully predict yielding under multiaxial stress conditions, there is no similar approach available to assess multiaxial fatigue with the same degree of accuracy. The use of yielding equivalent stress criteria, such as maximum principal stress or von Mises, for multiaxial fatigue life assessment is still quite spread and it is recommended by regulation codes [51]. However, the limits of this approach are widely known and large safety factors are prescribed in order to account for it’s lack of reliability.
Another class of approaches for multiaxial fatigue is represented by the so called critical plane approaches. In this case, instead of defining an equivalent value the stress (or strain) is evaluated only in a specific direction where the crack is supposed to develop.
2.2.5
Non-proportional loads
A common practice in fatigue testing is to apply loads with a sinusoidal waveform since this kind of signal can be easily implemented in the any testing machine controller and it is suitable for dynamic loading. In order to investigate the effects of multiaxial loading a number of experiments have been carried out where two independent loads are applied. In this case, it is generally possible to set any desired phase angle between the two load signals. This kind of loading is referred to as non-proportional and it has been found to produce relevant effects on the fatigue endurance of both plain and welded components. Additional information to this topic can be found in [52–55], just to cite a few of the many work concerning this subject.
Non-proportionality factor
For the classification to real case loadings where the waveform is far from being sinusoidal, a fatigue related mathematical definition of non-proportionality is needed and it has to be related to the effects on fatigue life produced by the non-proportionality itself. Particularly relevant aspects to this matter are the strain dependence of fatigue endurance shown by ductile materials (particularly at higher loads) and the non-proportional strain hardening behaviour shown by many metallic materials. Those reasons lead to a definition of a non-proportional factor based on the strain history. Therefore, loading conditions that produce a variation of the principal strain directions over time are referred to as a non-proportional. A number of factors have been defined in order to quantify the
degree of non-proportionality related to a load history. Among them, we can count the non-proportionality factor proposed by Kanazawa (Fp) [56], that can be expressed as the ration between the shear strain range acting over a plane oriented at 45◦ with respect to the maximum shear plane (∆γ45◦) and the
maximum shear strain range (∆γmax). The Kanazawa’s non-proportionality factor is expressed by Eq. 2.7. Since ∆γ45◦ has to be non-negative and lower or at
least equal to ∆γmax, we can obtain a variation range from zero to unity for the non-proportionality factor (0 ≤ Fp ≤ 1).
Fp =
∆γ45◦
∆γmax
(2.7) A better understanding of the non-proportionality factor physical meaning can be achieved looking at the strain paths generated from in-phase and out-of-phase tension-torsion load histories, as described by Socie and Marquis in [23]. Some explicative graphical representations of such strain paths is reported in Fig. 2.4. The strain history can be reproduced on a graph where the shear strain is plotted against the normal strain, those can be γxy and x respectively for our tension-torsion case where the axial load and the twisting moment are applied in the "x" direction.
Figure 2.4a represents the in phase loading, in this case the strain path is reduced to a straight line since the normal and shear strains are directly proportional to each other. In the example shown, the strain path is oriented as the bisector of the first and third quadrants, however in general the orientation is related to the selected ratio between axial and torsion loads. If the axial to torsion ratio is kept constant but the phase angle is increased to 90◦ an out-of-phase load is obtained and the strain path describes a circumference as reported in figure 2.4c. The latter is considered as the most damaging load condition (with regard of proportionality) since it produces the maximum amount of non-proportional strain hardening [23]. The non-proportionality factor reaches an unitary value (Fp = 1) for the described out-of-phase loading and it is equal to zero (Fp = 0) for in-phase loading, therefore those two conditions represent the extremes of its variation range. Any general load history (Fig. 2.4b) has to show a strain path with an intermediate behaviour between the two described above.
γ p 3 ² Proportional loading, Fp=0(b =0) (a) γ p 3 ² General loading (b) γ p 3 ² 90◦ out-of-phase, Fp=1(b =a) (c)
Figure 2.4: Non proportionality factor related to various load
A possible graphical interpretation of the non-proportionality factor Fg is related to the ration between the smaller and larger semi-axes of the minimum
ellipse enclosing the strain path in the von Mises diagram (Eq. 2.8) [57]. This definition results as an approximate evaluation of the Fp factor for general loading histories since the ellipse’s semi-axes (Fig. 2.4b) are an approximate description of the maximum shear range and of the shear strain range acting over a plane oriented at 45◦ with respect to the maximum shear plane that appear in Eq. 2.7. However, the two definition (Fp and Fg) lead to identical values if normal and shear strains are applied with a sinusoidal waveform and any phase angle since in this case the strain path shows an elliptical shape.
Fg = b/a (2.8)
The mathematical procedure of finding the smallest area ellipse that encloses a set of points is a classical mathematical problem. Here, the minimum volume ellipse has been obtained through a Matlab routine based on the optimizationR procedure proposed in [58].
2.2.6
Variable amplitude loads
The fatigue life assessment of components subject to variable amplitude loads is a matter of wide practical interest. In facts, the vast majority of structural components experience a service load history that is variable with time. Even in the few cases where the loading action can assumed to be rather constant, a limited number of overloads or underloads have to be accounted for. In facts, any component has to endure a certain degree of anomalous load conditions that can derive from improper use or malfunctioning.
The fatigue life assessment under variable amplitude load is a field of engineering where many questions are still left for an answer. Primarily because the high costs related to variable amplitude load testing prevents for its diffusion. If an elastic material model is assumed the loading order has no effect on stress-strain response of the material. This observation leaded to the formulation of the cumulative linear damage rule (LDR) from Palmgren and Miner [8]. LDR evaluation is made through equation 2.9, it postulates that the damages from load cycles at a specific load amplitude can be expressed as the ratio between the applied and the endurable number of cycles. Then, the total damage is computed as the linear summation of the partial damages related at each applied load amplitude cycle.
D =XDi =
X ni Ni
(2.9) The LDR is the most spread damage cumulation law, however, it’s reliability has been questioned many times and a number of non-linear damage rules has been proposed [59]. The main critic to LDR is that plasticity induced non-linearities of the material can not be taken into account by a linear damage summation that disregards loading sequence effects.
2.3
Estimation of local elastic-plastic stress-strain
at a notch
A geometric discontinuity in a component, like a notch, produces a stress concentration when a load is applied. The higher local stress and strains generally becomes the points where fatigue damage is more likely to manifest. For this reason there is a wide interest, and literature coverage, on the topic.
Nowadays the use of finite element techniques would seem the most direct and practical way to find a good approximation of the stress strain history at the notch tip during cyclic loading. Such an approach, however, involves a huge deal of computational effort especially for long, in terms of time, variable amplitude load histories, since the problem is non-linear and time dependent.
Another kind of approach is related to the use of empiric approximations. This can be a challenging task when multi-axial non proportional loads are involved, but offers two main advantages:
• time saving, since the empiric approximations are based on algebraic equations their solution is much more "lightweight" compared to non-linear FE models. This aspect is particularly relevant if complex time-dependant load histories are taken into account;
• understanding, the mental work implied in the formulation of a realistic empirical approximation leads to a better understanding of the multi-axial straining phenomena and then of the fatigue damage process.
2.3.1
Empirical approximations
Many empirical approximations have been suggested in order to estimate the elasto-plastic stress-strain field over a notch starting from its idealized elastic formulation. In the next paragraphs, two of the most spread ones will be reviewed: the Neuber rule [60] and the equivalent strain energy density [61].
Said analytical approaches share the common hypothesis of small scale yielding. Plastic deformation is assumed to take place only in a small material volume at the notch tip, while a preponderant portion of the component is supposed to show a linear elastic behaviour. This happens to be a quite spread condition for fatigue loaded notched components since the stress concentration due to a notch produces a linear stress distribution with high maximum value and steep gradients even when limited nominal stresses are applied. For higher loads the small scale yielding hypothesis would not result appropriate, however, in those conditions a very limited fatigue life is expected.
Neuber Rule (NR)
This rule was proposed by Neuber [60] and it has been widely used in order to evaluate the plastic non linear stress-strain behaviour of notched components in several fatigue life assessment procedures [62–64].
In its classical uni-axial formulation the rule states that the theoretical stress concentration factor is equal to the geometric mean of the actual stress and strain concentration factors:
Kt2 = KσK. (2.10) Equation 2.10 can be brought in the form of eq. 2.11 substituting at each concentration factor its definition.
∆σ∆ = Kt2∆σenom∆enom. (2.11) The described formulation allows to calculate the actual stress and strain acting over the notch (∆σ and ∆) starting from the nominal stress and strain values (∆σe
nom and ∆enom). However, in many fatigue related problems it is more convenient to adopt an equivalent formulation based on the local elastic stresses and strains (∆σe and ∆e), or rather the stresses and strains over the notch computed by means of a linear elastic material model. Those values can be easily obtained by linear FE analysis.
∆σ∆ = ∆σe∆e. (2.12) Equation 2.12 presents two unknowns (∆σ and ∆). The solution can be obtained introducing the material stress-strain relationship. The elastic strains can be easily related to the elastic stresses by the Hook’s law, while a Ramberg-Osgood relation [28] (here reviewed in sec. 2.1.1), which is generally adopted to describe the cyclic plastic behaviour on the left side of eq. 2.12.
True strain
(
²
)
Tr
ue
st
re
ss
(
σ
)
∆²
2=
∆2E
σ
+
³∆σ
2K
0 ´1 n0 Linear elastic Ramberg-Osgood Neuber's ruleFigure 2.5: Graphical representation of the Neuber’s rule
Graphically the NR can be represented as an hyperbole arc (dashed line) in the stress-strain diagram as shown in figure 2.5. Starting from the elastic solution (∆σe and ∆e) it is possible to obtain the desired inelastic values (∆σ and ∆) located at the intersection between the Neuber’s hyperbole and the Ramberg-Osgood curve. The described formulation is widely used in order to avoid the high computational cost required by the implementation of non-linear material’s models in the FE tools.
Equivalent strain energy density (ESED)
The equivalent strain energy density (ESED) approach was proposed by Glinka [61] in order to evaluate the inelastic stresses and strains near notches and cracks. The ESED is based on the assumption that the strain energy remains constant going from linear-elastic to inelastic material models. This concept can be formally expressed as per equation 2.13
Z ∆e 0 ∆σed∆e= Z ∆ 0 ∆σ d∆ (2.13)
True strain
(
²
)
Tr
ue
st
re
ss
(
σ
)
Linear elastic Ramberg-OsgoodFigure 2.6: Graphical representation of the equivalent strain energy density rule. From a graphical point of view ESED implies that the area under the linear elastic curve has to be equal to the area contained below the inelastic stress-strain curve. This concept is represented in the graph of figure 2.6 where a Ramberg-Osgood relation is used to represent the inelastic stress-strain relationship.
Multiaxial extension of NR and ESED
Both the NR and ESED are traditionally applied to circular notches in plane stress, loaded in bending or normal load. Said conditions are, in facts, equivalent to a uniaxial stress state since the actions on the free surface of the notch radius have to be equal to zero.
An additional set of equations is needed in order to extend the application of those analytical formulation to the case of multiaxial loading. To this point several proposals have been formulated in literature.
Hoffman and Seeger [65] proposed one of the fist multiaxial extensions for the NR. Their work was later developed by Köttgen et al. [66] for their pseudo stresses and strain formulation.
Moftakhar et al., instead presented a multiaxial extension for the ESED and compared the obtained results with the NR multiaxial formulation [67].
Comparison of NR and ESED
It is important to notice that both the NR and the ESED are formulated based on the hypothesis of small scale yielding, meaning that the error introduced is expected to increase with increasing plastic deformation.
Also both rules are originally formulated in terms of equivalent stresses and strains, therefore they the both need an additional set of equations in order to evaluate multiaxial load conditions.
It terms of expected results it can be stated that NR provides an upper bound estimation of notch root strain while ESED gives a lower bound estimation [67, 68]. A direct comparison of NR and ESED can be expressed for the uniaxial case under the hypothesis that the inelastic stress-strain relationship can be represented by a Ramberg-Osgood relation. Under said hypothesis the NR formulation from equation 2.12 can be brought in the form stated by equation 2.14 by substituting the strain terms on each side with their expression in terms of stress according to the elastic and inelastic stress-strain relationship respectively.
(∆σe)2 E = (∆σ)2 E + 2∆σ ∆σ 2K0 1/n0 , according to NR (2.14) If the same substitutions are made in the ESED formulation (eq. 2.13), the expression for the elastic stress range varies according to eq. 2.15.
(∆σe)2 E = (∆σ)2 E + 4∆σ n0+ 1 ∆σ 2K0 1/n0 , according to ESED (2.15) The equations relating elastic and inelastic stresses in the two cased differ in the term n02+1. Since the strain hardening coefficient is limited in the interval 0 < n0 < 1
we can state that n02+1 > 1. This implies that inelastic stress evaluated according
to NR is greater that the same value computed following the ESED formulation. Also, the NR strains have to be greater than the ESED ones since the inelastic stress-strain relationship is monotonic.
According to [23] NR provides more accurate stress-strain assessment in plane stress conditions, while ESED results to be more accurate in plane strain conditions. However, in order to obtain a precautionary approach in stress-strain estimation the use of NR has to be preferred to ESED since NR always predicts greater stresses and strains compared to ESED.
2.3.2
Analytical formulations
Starting from the early work of Inglis [69], who analysed the stress distribution due to elliptical holes in wide plates, many researchers have dealt with the analytical description of the linear elastic stress-strain state surrounding a notch. Among them, Neuber [10] proposed his classical approach, based on Airy’s biharmonic potential function [70], to determine the stress concentration factors of bluntly notched components. Later on, Westergaard [71] and Muskhelishvili [72] formulated a more general theory, where a complex representation of Airy’s
