BiaxiaVMultiaxial Fatigue and Fracture
Andrea Carpinteri, Manuel de Freitas and Andrea Spagnoli (Eds.)
0 Elsevier Science Ltd. and ESIS. All rights reserved. 463
FATIGUE ASSESSMENT OF MECHANICAL COMPONENTS UNDER COMPLEX MULTIAXIAL LOADING
JosC L.T. SANTOS, M. de FREITAS, B. LI and T.P. TRIG0 Dept. of Mechanical Engineering, Instituto Superior Tkcnico
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
ABSTRACT
This paper addresses an integrated FEM based approach for crack initiation life assessment of components under complex multiaxial loading. Generally, there are many sources of error in the computational fatigue damage assessments, including uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc. This paper is focused in the methodology for handling the effect of non-proportional multiaxial loading, and in improvements in computational algorithms for reducing the computation time for fatigue assessments. Since the effective shear stress amplitude is an important parameter for crack initiation life prediction, the recent approaches on evaluating the effective shear stress amplitude under comlex loading paths are studied and compared by examples. The MCE approach developed on the basis of the MCC approach is described in detail, and it is shown that this approach can be easily implemented as a post-processing step within a commercial FEM code such as ANSYS. Fatigue assessments of two application examples are shown, using the computational procedure developed in this research. The predicted fatigue damage contours are compared for proportional and non-proportional loading cases, it is concluded that the fatigue critical zone and fatigue damage indicator vary with the combined conditions of multiaxial fatigue loading. Advanced multiaxial fatigue approaches must be applied for fatigue assessments of components/structures under complex multiaxial loading conditions, to avoid unsafe design obtained from the conventional approaches based on the static criteria.
KEYWORDS
Multiaxial fatigue, fatigue damage evaluation, computational durability assessment, fatigue life prediction.
LNTRODUCTION
Due to the increasing pressure of market competition for light weight design and fuel economy, computational durability analysis of engineering components/structures is more and more used in today’s industrial design for reducing prototype testing and shortening the product development cycle [ 1 I. Since it is widely recognized that about 80% of mechanical/structural component failures are related to fatigue, structural fatigue life has become the primary concern in design for durability.
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In real service, engineering components and structures are generally subjected to multiaxial fatigue loading conditions, in which the cyclic loads act in various directions, with different frequencies and/or different phases [2]. In these non-proportional multiaxial loading conditions, the corresponding principal directions and/or principal stress ratios vary during a loading cycle or block. Advanced engineering designs require efficient, accurate and easy-of- use methods for durability assessment of components/structures under complex multiaxial loading.
Current fatigue design approaches treat both proportional and non-proportional loading with the maximum principal or equivalent stress range, and then, they refer to the design S-N curve obtained under uniaxial loading condition [3]. The Eurocode 3 design code recommends that the maximum principal stress range may be used as a fatigue life damage parameter if the loading is proportional. For non-proportional loading, the components of damage for normal and shear stresses are assessed separately using the Palmgren-Miner rule and then combined using an interaction equation. Maximum shear stress range is used as an equivalent stress for non-proportional loading in the ASME code.
However, conventional multiaxial fatigue criteria were based on proportional fatigue data, and hence not applicable to non-proportional loading, due to the changes in direction and/or ratio of the principal stresses. This has led to a number of research studies on the multiaxial fatigue problem over the past 20 years. Much progress has been made in understanding the cracking modes under complex loading, and various multiaxial fatigue damage parameters have been proposed.
Although many multiaxial fatigue models have been proposed in the literature, there still exist gaps between the theoretical models and engineering applications. Generally, there are many sources of error in the computational fatigue damage assessments, including uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc. It is imperative to study the accuracy and improve the computational algorithms for every step of the fatigue evaluation process.
The objective of this paper is to study the engineering approaches for crack initiation life assessment of components under complex multiaxial loading. Firstly, current multiaxial fatigue models are briefly reviewed and compared. Then the recent approaches for evaluating the effective shear stress amplitude under complex loading paths are studied and compared with example problems. It is shown that the minimum circumscribed ellipse (MCE) approach, developed on the basis of the minimum circumscribed circle (MCC) approach, is an easy and efficient way to take into account of the non-proportional loading effect for fatigue evaluations.
The stress invariants based multiaxial criterion, coupled with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude, are shown to be a simple and efficient methodology for handling the complex loading effects.
The implementation of the minimum circumscribed ellipse (MCE) approach in the commercial FEM code ANSYS is discussed. Applications of the developed procedure for engineering problems are shown for two examples: an automotive suspension torque arm, and a train car.
In the integrated FEM based fatigue assessment procedure, the quasi-static FE analyses are used to obtain the stress-time histories at each nodal point by stress superimposition due to each individually applied load. Then the minimum circumscribed ellipse (MCE) approach is used for multiaxial fatigue life evaluation at each nodal point, requiring only the knowledge of basic material fatigue parameters.
Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading 465
CURRENT METHODOLOGIES FOR FATIGUE DAMAGE EVALUATIONS
The fatigue life of a mechanical component or structure depends on the interaction of at least three physical and mechanical phenomena: the material behaviour, the geometry of the component, and the service loading of the component or structure [I].
The fatigue damage assessment methods can be categorized as two groups: global approach and local approach [4]. The global approach uses directly the amplitudes of the nominal stresses or the acting forces/moments, and compares them with the nominal stress S-N curve for fatigue limit evaluation or fatigue life prediction. The local approach evolved from the global approaches, and proceeds from local stress and strain parameters, consists of different types: structural stress approach, notch root approach, and so on.
The structural stress approach proceeds from the structural stress amplitudes in the component/structure, and compares them with a structural stress S-N curve. The structural stresses (also called hot spot stresses) are generally the results of finite element analysis of welded or nonwelded structures, without consideration of the actual notches (such as the welding geometry, etc.) in the finite element modelling. Commonly, the structural stresses are elastic and indicate the macro-geometrical influences.
The notch root approach proceeds from the elastic-plastic strain amplitudes at the notch root and compares them with the strain S-N curve of the material in the unnotched comparison specimen. The notch root approach is also called the local strain approach, and is based on the hypothesis that the mechanical behaviour of the material at the notch root in respect of local deformation, local damage and crack initiation is similar to the behaviour of a miniaturized, axially loaded, unnotched specimen in respect of global deformation, global damage and complete fracture.
Different views exist between experts concerning how detailed the local consideration must be in the fatigue assessment procedure, based on structural stresses only or on notch stresses also. No general answer is possible. The choice of the approach must be made based on the circumstances of the case considered.
Generally, the structural stress analysis is always required because the notch stresses/strains are based on structural stresses. If the scatter range of the local notch geometry, caused by the manufacturing process, is small or if the scatter range can be passed over by a worst-case consideration, the step from the structural stress approach to the notch stress approach is justified. However, if the scattering of the notch geometry is very significant such as the case of non-machined welded joints, the notch stress analysis is not well suited because the notch geometry cannot be accurately modelled.
Due to the complex geometry of engineering components and structures, the nominal stresses cannot meaningfully be defined. The local approach is widely used in the computational fatigue assessment procedures, which involves isolating each potential critical location and independently determining its fatigue life. By isolating each potential fatigue critical location, the complex component is regarded as a number of individual fatigue specimens. The most fatigue-critical location is then the location with the shortest fatigue crack formation life. The fatigue life of the component is therefore defined by the fatigue life of the most fatigue-critical location.
Computer aided fatigue evaluation of engineering components/structures consists of two main steps: dynamic stress computation and fatigue life prediction. Dynamic stress histories can be obtained either from experiments (mounting sensors or transducers on a physical component) or from computer simulation. The simulation-based approach is usually done by performing finite element analysis of the component under the specified set of applied loads.
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Then, fatigue life prediction is carried out as a post-processing step of finite element output results. A general flow chart of computational fatigue assessment is shown in Fig. 1.
Analysis Element Analysis
Load Time Histories
I . Static Stresses for Unit Loads
Principle
a l
I Stress Time Histories I
a
ultiaxial Fatigue Criteria
Fatigue Life Prediction
Fig. 1. Schematic flowchart of computational fatigue assessment.
Uncertainties in computational fatigue assessments are attributable to many sources, such as uncertainties in analysing complex service environments, complex geometries, and lack of usable material information, etc.
Among the many sources of errors in the computational fatigue assessment, the effect of non-proportional multiaxial loading is one of the important considerations, since recent researches have shown that the non-proportional loading causes additional fatigue damage and the conventional methodologies of multiaxial fatigue life assessment may lead to unsafe design.
EVOLUTION OF MULTIAXIAL FATIGUE PREDICTION METHODS
The multiaxial fatigue criteria proposed in the literature may be categorized in three groups:
stress-based, strain-based and energy-based methods. For high-cycle fatigue problems, most of the multiaxial fatigue criteria are stress-based. Early works on multiaxial fatigue include the extension of the von Mises criterion to the S-N curve, which has been widely used for proportional cyclic stresses where ratios of principal stresses and their directions remain fixed during cycling.
Fatigue Assessment of Mechanical Components Under Complex Multiaxial hading 467
In order to handle non-proportional loading effect on fatigue resistance, many new methodologies have been developed and are based on various concepts such as the critical plane approach [ 5 ] , integral approach [6], mesoscopic scale approach [7], etc.
A common feature of many high-cycle multiaxial fatigue criteria is that they are expressed as a general form and include both shear stress amplitude r, and normal stress u during a loading cycle:
T, + k ( N ) o = R ( N ) (1)
where k(N) and h(N) are material parameters for a given cyclic life N . Multiaxial fatigue models differ in the interpretation of how shear stress and normal stress terms in Eq. (1) are defined.
For non-proportional cases, a stress-based version of the ASME boiler and pressure vessel code, case N-47-23 [ 8 ] , may be used as an extension of the von Mises criterion, in which an equivalent stress amplitude parameter, SEQA, is defined from stress ranges Abx, Aby, Abz, Arxy, Azyz. AT^,, in the form
ScQa = x J A c , I - A e , ) 2 +(ACT, - A c i ) 2 + ( A c i - A c , ) ' +6(Arm2 AT,:^ + A T , ' ) (2) where Ao,=o,(t,)-o,(tz), Ao,=o,(t,)-o,(tz), etc. SEQA is maximized with respect to two time instants, 11 and t2, during a fatigue loading cycle.
For constant amplitude bending and torsional stresses such as
Eq.(2) becomes
3 9
S, =%/I + K 2 +,/I +: I( cos(2Sx,) + - 16 K . (4) where K=2ztla,.
When r,/ob=0.5 and 6,,=0 (proportional loading case), Eq. (4) gives S,, = 1.3230,. When z,/ob=0.5 and F,,=90" (out-of-phase loading case), Eq. (4) gives S,, = b h , which means that out-of-phase load case is predicted to be less damaging than the proportional load case with the Same stress amplitudes.
However, experimental results showed that the prediction by Eq. (4) for out-of-phase load case is inconsistent and non-conservative. Hitherto, many approaches have been proposed for treating the non-proportional effects, among them the critical plane approach and the integral approach are two important concepts.
Critical Plane Approaches
Critical plane approaches are based upon the physical observation that fatigue cracks initiate and grow on certain material planes. The orientation of the critical plane is commonly defined as the plane with maximum shear stress amplitude. The linear combination of the shear stress
468 1L.Z SANTOS E T AL.'
amplitude on the critical plane and the normal stress acting on that plane is defined as the fatigue damage correlation parameter.
For complex loading histories, the principal directions may rotate during a loading cycle (e.g. see Ref. [9]). Therefore, Bannantine and Socie [5] suggested that the critical plane should be identified as the plane experiencing the maximum damage, and the fatigue life of the component is estimated from the damage calculations on this plane. The approach proposed by Bannantine and Socie [5] defines the critical plane as the plane of maximum damage rather than the plane of maximum shear stress (strain) amplitude, as defined by previous authors. This approach evaluates the damage parameter on each material plane. The plane with the greatest fatigue damage is the critical plane, by definition. For general random loading conditions, with six independent stress components, the critical plane approaches have to be carried out for plane angles 8 and Q varying from 0 to E . These procedures demand a great deal of calculations, especially when small angle increments are used.
In the last decades, the critical plane approaches have found wide applications and also received some criticism. The critical plane approach assumes that only the stress (strain) acting on a fixed plane is effective to induce damage, and then, no interaction of the damages on the different planes occurs. These assumptions are not always valid, and may considerably underestimate fatigue damage. Zenner et al. [lo] also indicated by a typical example that the hypotheses of the critical plane approach are not suitable for describing the effect of the phase difference. The example considers the stress waves of Eq. (3), with phase shift angle Sxy= 90' and stress amplitude q= 0.5ab. Under this load case, the shear stress amplitude has the same magnitude in all planes.
Integral Approaches
Integral approaches are based on the Novoshilov's integration formulation, as a mean square value of the shear stresses for all planes [lo]:
Equivalent-stress amplitude is yielded by an integration of the square of the shear stress amplitude over all planes y+ for fully reversed stresses.
Further developments of the integral approaches led to various hypotheses such as the effective shear stress hypothesis, the shear stress intensity hypothesis (SEI), etc. Generally, the integral approach [lo] uses the average measure of the fatigue damage by integrating the damage over all the planes. The integral approach considers all damaged planes of a specific critical volume. The averaged stress amplitude of the shear stress intensity hypothesis (SEI) is formulated as:
Papadopoulos' mesoscopic approach 11 11 is also formulated as an average measure, by integration of the plastic strains accumulated in all the crystals, within the elementary volume:
Fatigue Assessment of Mechanical Components Under Complex Multiaxial Loading 469
where the angle ty, varying from 0 to 2 x , covers all the gliding directions on a material plane, whereas the angles cp and y, varying from 0 to 2 x and from 0 to IC respectivery, cover all the possible material planes.
These procedures also demand a great deal of calculations, especially when small angle increments are used.
A recent comparison study by Potter et al. 161 showed that the integral approach is superior to the critical plane approach under varying principal stress directions.
Stress Invariants Based Approaches
The von Mises criterion has always been of vital importance in establishing strength hypotheses for cyclic load. Based on extension of the von Mises criterion and on inclusion of the hydrostatic stress effect, many multiaxial HCF criteria are formulated in the forms of stress invariants. Among them, the Sines [ 121 or Crossland [ 131 criteria are two important criteria, which are formulated by the amplitude of the second deviatoric stress invariant and the hydrostatic stress PH
Crossland suggested the use of the maximum value of the hydrostatic stress PH,- instead of the mean value of hydrostatic stress P H , ~ proposed by Sines. A physical interpretation of the criterion expressed in Eq.(8) is that for a given cyclic life N, the permissible amplitude of the root-mean-square of the shear stress over all planes is a linear function of the normal stress averaged over all planes.
From the viewpoint of computational efficiency, the stress-invariant based approach (see Eq.(8)) is easy to use and computationally efficient.
In practical engineering design, the Sines [12] and Crossland [ 131 criteria have found successful applications for proportional multiaxial loading. For non-proportional multiaxial loading, the Sines and Crossland criteria can also yield better prediction results by using improved method for evaluating the effective shear stress amplitude of the non-proportional loading path as shown in [14, 151. This will be discussed in the following section.
EFFECTIVE SHEAR STRESS AMPLITUDE UNDER COMPLEX MULTIAXIAL LOADING
For multiaxial fatigue analysis using Eq.(8), it is an essential task to compute the values of (representing the shear stress amplitude) and P H (hydrostatic stress) during a loading cycle or block. Since the computation of the hydrostatic stress PH is easier, it is not discussed here and only the computation of 6 is addressed in this section.
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Approaches for Evaluating Shear Stress Amplitude under Multiaxial Loading The definition of the square root of the second invariant of the stress deviator is:
&&{(rXx 1 -6,,)2 +(e,, -crr)2 +(6, -6,)* +6(c,,' + e r r 2 ] ( 9 )
One direct way to calculate the amplitude of is:
Eq.(lO) is applicable for proportional loading, where all the stress components vary proportionally. However, when the stress components vary non-proportionally (for example, with phase shift between the stress components), Eq.(lO) gives the same result with that of proportional loading condition. In fact, the non-proportionality has influence on the shear stress amplitude generated by multiaxial loading. Therefore, a new methodology is needed.
The longest chord approach is one of the well-known approaches as summarized by Papadopoulos in [16], which defines the shear stress amplitude as half of the longest chord of the loading path, denoted as D/2. Based on the longest chord approach, an improvement was proposed to provide a detailed characterization of the loading path by Deperrois et al in [ 171.
For combined bending and torsion fatigue problem, the application of Deperrois approach can be described as: firstly find the maximum chord of the loading path curve, denoted as 4, and then project the loading path curve onto the line perpendicular to D2. This projection is a line segment with the length equal to D I . The shear stress amplitude is defined as !-,/=.
A weakness of the Deperrois approach is the breakdown for cases with non-unique maximum chord. Then a multitude of lines exist, each one perpendicular to a different maximum chord oy) i = 1,2 ... , consequently a multitude of projections exist, inducing the breakdown of Deperrois approach.
To overcome the inconsistency of the longest chord approach, the minimum circumscribed circle (MCC) approach was developed by Dang Van [7] and Papadopoulos [16]. The MCC approach defines the shear stress amplitude as the radius of the minimum circle circumscribing to the loading path. However, the minimum circumscribed circle (MCC) approach cannot differentiate proportional loading path from non-proportional loading path, which means that the MCC approach cannot characterize the non-proportional loading effect.
On the basis of the MCC approach [7, 161, a new approach, called the minimum circumscribed ellipse (MCE) approach [14, 151, was proposed to compute the effective shear stress amplitude taking into account the non-proportional loading effect.
The load traces are represented and analysed in the transformed deviatoric stress space [ 181, where each point represents a value of f i and the variations of are shown during a loading cycle. The schematic representation of the minimum circumscribed ellipse (MCE) approach and the relation with the minimum circumscribed circle (MCC) approach are illustrated i n Fig. 2.
2