BiaxiaVMultiaxial Fatigue and Fracture
Andrea Carpinten, Manuel de Freitas and Andrea Spagnoli (Eds.)
Q Elsevier Science Ltd. and ESIS. All rights reserved. 383
A MULTIAXIAL FATIGUE LIFE CRITERION FOR NONSYMMETRICAL AND NON-PROPORTIONAL ELASTO-PLASTIC DEFORMATION
Mauro FJLIPPINI' , Stefan0 FOLETTI',
Ioannis V. PAPADOPOULOS2 and Cetin Morris SONSIN03
' Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy European Commission, JRC, IPSC, Ispra, Italy
' Fraunhofer-Institute for Structural Durability LBF, Darmstadt, Germany
2
ABSTRACT
A new low-cycle multiaxial fatigue life prediction methodology based on the concept of an effective shear strain is proposed. This effective shear strain is derived by averaging the total shear strains acting on all planes passing through a material point. The proposed model, which is formulated as a generalised equivalent strain, takes into account the effect of non- symmetrical loading cycles. The main advantage of the model relies on the small number of material parameters to be identified. The axial cyclic stress-strain curve, the basic strain-life curve (Manson-Coffin) and an additional life curve obtained under zero to tension strain controlled axial fatigue tests are sufficient to allow application of the proposed criterion in all loading conditions. The experimentally observed fatigue lives of proportional and non- proportional multiaxial strain controlled low-cycle fatigue tests from un-notched tubular specimens, have been compared with the predicted lives of the proposed approach showing in all cases a good agreement.
KEYWORDS
Multiaxial fatigue criteria, strain-controlled fatigue, mean strain, Inconel 7 18 alloy, steel.
INTRODUCTION
Since many mechanical components are subject to cyclic multiaxial loading, fatigue evaluation is becoming one of the major issues in the lightweight design of structures. Many methods have been proposed to reduce the complex multiaxial stresdstrain state to an equivalent uniaxial condition, namely empirical formulas, stress or strain invariants, strain energy, critical plane approaches and space average of stress or strain. Historically, the first multiaxial low- cycle fatigue criteria have been based on the extension of static criteria, e.g. maximum principal strain, maximum shear strain or maximum octahedral shear strain criteria: the main
3 84 M. FILIPPINI ET A t .
disadvantage of these criteria is that their application is limited to the case of fixed principal stress or strain directions during the loading cycle. Modified versions of such criteria, so that application to out-of-phase loadings is made possible, have been also proposed [ 13.
In the so-called critical plane approaches, quantities related to the mechanism of formation of fatigue cracks under multiaxial loading are inserted explicitly in the formulation of the criteria: a combination of normal and shear stresses or strains acting on particularly oriented planes, on which fatigue cracks are likely to nucleate, is chosen as the critical parameter for assessing the fatigue life of components submitted to multiaxial cyclic loading. Among critical plane approaches, a distinction between criteria formulated in terms of strain or in terms of both stress and strain is also possible. Following the proposal of Brown and Miller [23 (r-
plane) and successive contributions [3f, the shear and the normal strain acting on the plane of maximum alternating shear strain are used. Though these criteria employ exclusively strain- related quantities, they should be classified in the category of critical plane approaches, rather than in the strain based criteria (see Socie and Marquis [ 5 ] ) . The proposals of Socie [6], where combinations of stress and strain acting on critical planes are used to predict fatigue life, have been applied for predicting fatigue behaviour in the intermediate life region. The critical plane approach is given a physical justification based on the observations of nucleation and early growth of fatigue cracks but, in most cases, its adoption is limited by the need of developing complex multiaxial material models.
The observation of hysteresis loops in low-cycle fatigue testing have suggested many authors the formulation of criteria based on the relationship between the total or the plastic energy in a loading cycle and the fatigue life. These criteria are usually grouped under the name of energy criteria: among many others, the proposals [7,8,9] may be considered.
However, the major obstacle to the application of criteria based on strain energy is either the necessity of the complete loading histories of all the components of stress and strain tensors or the availability of a material model able to reproduce the stress-strain loading paths experienced by the material. More detailed review of multiaxial fatigue criteria can be found in references [5,10,11,12].
In this paper a new approach based on a space average of the tensor of total strains reducing the complex loading history to an effective equivalent strain is presented. The proposed approach, based on an extension of the Sonsino-Grubisic methodology [13], takes into account the effect of shear strains on crack initiation, expanding the investigation of the interaction of shear strains on all different interference planes. This new approach makes possible to link the advantages of a strain based criterion with the possibility of taking into account the different material behaviour due to out-of-phase loads and the modifying effect of superimposed mean strains. In general, the advantage of criteria based on total strain is that they may be easily applied without making use of an elasto-plastic multiaxial model, at least in the case of simple components or specimens. In the case of complex geometry structures, the strains at the critical points have still to be calculated by means of finite element method in combination with a suitable material model. Alternatively, measured strains by means of strain gauges may be employed in combination with the criterion presented in this paper for predicting the fatigue life of a component.
Moreover, the possibility of taking into account the effect of a mean strain allows extending the use of the new criterion to the range of intermediate fatigue life (about lo5 cycles). The effect of mean strains on the fatigue life may be neglected in the low-cycle fatigue range;
nevertheless it may seriously affect fatigue life in the intermediate life range up to the high- cycle fatigue regime. This effect is more evident in the case of superalloys and hard metals, where the mean strains are closely related to the mean stresses, even at shorter lives.
A Multiaxial Fatigue Life Criierion for Non-Symmetrical and Non-Proportional Elasto-Plastic ... 385
A SELECTIVE REVIEW OF STRAIN-BASED CRITERIA
Many mechanical components and structures are often subject to complex elasto-plastic strain states, particularly at stress concentration zones such as notches. For a uniaxial stress state, in the low and intermediate range of life, a fatigue life prediction may be obtained by the Manson-Coffin equation:
6’
Ea =-1(2N,)b+&;(2NJ E
where 0; and b are the fatigue strength coefficient and exponent respectively, E; and c are the fatigue ductility coefficient and exponent respectively, and E is the Young modulus. Clearly, the above relation is not able to take into account the effect of multiaxial loading. In the last years different multiaxial fatigue life prediction methods have been proposed [IO] for assessing the fatigue life under complex loads. Strain-based criteria are obtained by casting a multiaxial strain state into an equivalent uniaxial strain. Some of the strain-based fatigue life prediction methodologies are briefly reviewed in the following.
von Mises criterion
One of the most common equivalent strain-based criteria is the maximum octahedral shear strain amplitude criterion. For a multiaxial strain state, this hypothesis defines an equivalent strain amplitude through the relationship:
where E,,,~ and x,,, denote respectively normal and shear strain amplitudes and v is the Poisson’s ratio. In the following, this criterion will be named after vonMises, even if the original proposal by von Mises, currently employed in plasticity for determining the onset of yielding, is based on the strain energy density of distortion. According to this approach, one obtains a fatigue life prediction replacing into the Manson-Coffin relationship the axial strain amplitude E, with the equivalent strain amplitude E,,,, , given by Eq. (2).
Let us consider two load states both having the same axial and shear strain amplitudes; in the first state the strains are in-phase whereas in the second they are out-of-phase. The major drawback resulting from the hypothesis of von Mises is that it produces the same equivalent strain for both the in-phase and out-of-phase load states above. Consequently, both states would result to the same fatigue life according to von Mises approach. Several experimental results contradict this prediction, showing that, for strain controlled fatigue tests, the fatigue life under out-of-phase loading is lower than the fatigue life under in-phase loading at the same applied strain amplitudes.
ASME Code
The ASME Boiler and Pressure Vessel Code Procedure [l] is based on the von Mises hypothesis. An equivalent strain range is defined through the relationship:
386 M. FILIPPINI ET AL.
+6 ( + A&: + A&;)] ' maximized with respect to rime
(3)
The terns A€, , Asq have to be calculated as strain differences between two generic instants tl and r2, e.g. A&, = E, (1, ) - E, (r, ) , A E ~ = sXy (t, ) - cXy (I, ) etc. The equivalent strain range Asq , Eq. (3), is calculated by varying tl and rz such as to obtain its maximum value. This criterion produces a lower equivalent strain for the out-of-phase than for the in-phase loading, predicting an increase of the fatigue life, in contradiction with the experimental results. The application of this criterion may lead to unconservative predictions, as shown by Tipton and Nelson [14].
Criterion of Sonsino and Grubisic
The criterion of Sonsino and Grubisic [I31 assumes that the fatigue damage is caused by the interaction of shear strains acting on different elementary material planes, called interference planes. An interference plane is completely defined by the spherical coordinates, 29 and p, of its unit normal vector n (Fig. 1).