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Development of a new simple energy method for life prediction in multiaxial fatigue
Accepted Manuscript Development of a new simple energy method for life prediction in multiaxial fatigue C. Braccesi, G. Morettini, F. Cianetti, M. Palmieri PII: DOI: Reference:
S0142-1123(18)30087-2 https://doi.org/10.1016/j.ijfatigue.2018.03.003 JIJF 4604
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
27 November 2017 2 March 2018 3 March 2018
Please cite this article as: Braccesi, C., Morettini, G., Cianetti, F., Palmieri, M., Development of a new simple energy method for life prediction in multiaxial fatigue, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/ j.ijfatigue.2018.03.003
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DEVELOPMENT OF A NEW SIMPLE ENERGY METHOD FOR LIFE PREDICTION IN MULTIAXIAL FATIGUE C. BRACCESI, G. MORETTINI(o), F. CIANETTI, M. PALMIERI University of Perugia, Department of Industrial Engineering, Via. G. Duranti 1/A-4 - 06125 Perugia – Italy (o) corresponding author, e-mail: [email protected], tel: +39 329 0049 720
Abstract. In this paper, a new, simple Damage Energy Criterium for life prediction in multiaxial fatigue, developed in the frequency domain, is described and verified. The first aim of this paper is to demonstrate that through a correct use of the energetic approach, developed in the frequency domain, it is possible to achieve an accurate result. The proposed theoretical procedure finds its conceptual foundation in the fact that the energy criterion cannot be directly applied to usual stress state in the time domain; however, with a simple passage it is possible to accurately reduce a multiaxial stress state to an equivalent uniaxial one. For this reason, this work will be characterized by analytical content and by substantial mathematical demonstrations. At the end, in order to verify the method and consolidate the general procedure, some experimental tests are used to corroborate the validity of this criterion. This proposal shows drastically simpler technique than any other proposed in current literature.
Keywords: multiaxial fatigue method; fatigue life prediction; energy multiaxial method; damage criteria; energy density criteria.
NOMENCLATURE Zeros matrix of -rows and -columns Compliance matrix Deviatoric matrix operator E
Young’s modulus Complex strain vector Deviatoric complex strain tensor Hydrostatic complex strain vector Cross Spectral Matrix of Cross Spectral Matrix of Hydrostatic matrix operator Identity matrix of -order Imaginary unit Fatigue limits ratio
ν
Poisson’s coefficient
Orthogonal matrix of QR decomposition of Upper deviatoric triangular matrix, QR decomposition of “real part” operator Upper hydrostatic triangular matrix Complex stress tensor Complex stress vector Deviatoric complex stress tensor Hydrostatic complex stress tensor Complex spectrum of alternating stress vector Deviatoric complex stress components Hydrostatic complex stress scalar value Complex stress components Complex spectrum of mean stress vector Tensile stress fatigue limit Ultimate tensile strength Yield tensile strength Alternating stress Eequivalent alternating sigma Stress components in time domain Mean stress t
Time Torsion stress fatigue limit Total energy density Deviatoric energy density Hydrostatic energy density
ω
Frequency
INTRODUCTION Multiaxial fatigue is a topic of great interest for industry and academic world. Therefore, the fatigue design of structural components is actually subject to considerable attention. In the scientific community, there is no universally accepted criteria regarding the methodology for the study of such a problem, although it is worth mentioning that in recent years, several methods have been presented in the literature. It is possible to classify these methods into different categories depending on the parameters and the approach with which the phenomenon is analysed. The first significant distinction stems from dividing the criteria into two categories: Safe/Unsafe Criteria and Damage Criteria. The first category determines whether the component, subjected to a multiaxial stress state, is resistant or not. To better analyse these criteria, it is useful to refer to their graphic interpretation. Indeed, each of these [1-3] defines one or more failure straight lines that split the plain, on which the loading path can be drawn, in a safety and unsafety domain [4]. A loading path that remains into bounding failure lines is expected to have infinite life while any path that extends outside the damage line will have fatigue failures. With this type of approach, only a safety factor for fatigue limit can be evaluated. F. Morel [5] was the first who tried to apply this approach to the damage evaluation concept. With his work, he tried to formulate a method to evaluate the damage. Through the second category, (Damage Criteria), it is possible to evaluate the damage caused by load time history as well as the damage contribution of a part of the stress signal. To this day, most multiaxial fatigue criteria fall under this category. Within the Damage Criteria it is possible to make a further distinction based on the approach with which the damage is calculated: the Geometric Approach and the Energetic Approach. The Geometric approach is also known as “Critical Plane” approach, which has been evolving since W. N. Findley [6] first proposed his critical plane-based criterion in 1959. This approach is based on the experimental observation that fatigue cracks initiate and grow on a certain material plane. Only stress and/or strain components acting on the critical plane are responsible for fatigue failure of the material. The chosen geometric reference system, and then of the critical plane, can happen either a priori or by iteration, using the maximization of some stress and/or strain parameters. According to Matake [7] the critical plane orientation equals the maximum shear stress amplitude, whereas according to Z. Q. Tao [8], it equals the maximum shear strain. For the other [9-14] this particular parameter could vary. The Critical Plane Damage criteria require calculation times which depend on the number of physical planes investigated during the fatigue limit estimation. It is clear that the fatigue limit is correlated to the micro-structure of the material [15] and thus to its geometry; however, the Energy Damage criteria are based on the assumption that it is possible to analyse the damage fatigue phenomenon using another parameter, the energy density, which is independent on the geometric reference system of choice. Energy density thus represents the only parameter necessary for the estimation of fatigue limit of metallic structural components subject to a multiaxial stress state. The first researcher to use an energy-based criterion for multiaxial fatigue was Y. S. Garud [16]. In 1981, he proposed an energy criterion associated with plastic deformation. Subsequently, F. Ellyin [17] proposed an approach based on the combination of elastic and plastic strain energy which demonstrates that these quantities, [18], result from the slipping of more than one plane. In the last decades, different Damage Energy criteria have been proposed in literature, such as the Park criterion or Susmel criterion [19-20]. The use of the Energy criteria presents a fundamental advantage when compared with the criteria discussed above: the energy quantity is independent on the component’s geometry. Through the use of the energy value alone, it is possible to evaluate the damage caused by load time history very quickly. As for the other categories of methods, Damage Energy Criteria can be classified into two categories that depend on the domain these methodologies have been developed in: the Time Domain and the Frequency Domain. Most of the times, loading conditions of a system are expressed in the Time Domain; these signals can thus be used without any pre-processing. The great advantage of this methodology is that there
are no limitations in terms of statistical distribution of the cycle. On the other hand, however, time stress histories are usually random, so in the time domain, even if it is possible to find the complete load history, it will be impossible it will be impossible to find load cycles and consequently an average energy value [17]. The energy expressed in this domain is a squared value and carries errors related to the doubling of the stress signal frequency. The authors affirm that this is the fundamental reason why the use of energy methods have come to be underestimated by the scientific community. The authors will show that solution to the above problems can be found in a procedure involving the use of Frequency Domain analysis. This subclass of methods holds that a simple elaboration of the signal is sufficient to correctly evaluate the energy of a given system. With this work, the authors propose a new simple Damage Energy Criterion for life prediction in multiaxial fatigue, developed in the Frequency Domain. The first aim of this paper is to demonstrate that through the correct use of the energetic approach, developed in the frequency domain, it is possible to achieve an accurate result. For this reason, this work will be characterized by both analytical content and substantial mathematical demonstrations. In particular, the first section of the paper introduces the stress tensor, its definition in the frequency domain and its relationship with the strain tensor. In the second section, the analysis focuses on the utility of dividing the stress tensor in two parts: the deviatoric and the hydrostatic one. In the third section it is then possible express the deviatoric energy density in the frequency domain. At the beginning of the fourth section all of the basic theories are introduced to describe the correct procedure to evaluate the equivalent alternating stress by an energetic approach. The fourth section is of paramount importance for this work; in fact, the most significant conclusions are expressed in this section by the authors. In order to complete the examination, the following chapters demonstrate that it is possible to extend the energy processing to eliminate errors stemming from the behavior of fatigue materials, as well as errors stemming from the influence of the mean stress. Such procedure is based on a precedent work [4] and presents a formal and substantial analogy with the von Mises theory [21]; indeed, for the special case of real static stress state, it gives the same result as the classical von Mises criterion. At the end of the paper, experimental validation is presented to confirm that the technique is drastically simpler than others proposed in literature but has a good accuracy. The simplicity of this method could assist engineers in verifying the fatigue limit of a structure stressed by multiaxial loads, especially during the initial design phase.
1. STRESS TENSOR, ITS DEFINITON AND ITS RELATIONSHIP WITH THE STRAIN TENSOR Consider a state of stress in a point described by a time varying stress tensor. Assume that the function that describes the variation of each stress component is sinusoidal with a particular frequency and a mean value of zero. By Euler’s formula [22], which establishes the fundamental relationship between trigonometric and complex exponential functions, it is possible to express each stress component of the time signal as real part of a complex quantity described by the following relation: (1)
where is the real part and is the imaginary part of the stress; these values can be considered as Fourier’s coefficients according to Fourier analysis of a time signal [23]. At this point, it is useful to focus on the complex representation of each stress component and generate the complex stress tensor (2), where each element is the complex spectrum of the corresponding deterministic time signals. This tensor represents the complex spectra for a particular frequency ω and it characterizes
the stress in a specific component point. Complex tensor components can be arranged in a complex spectral vector , as shown below: (2)
The symbol ' refers to the transpose operation. For the sake of simplicity, it could be useful to reduce, the complex stress vector in another one that only contains six complex components: (3)
In order to complete our examination, and to better analyse the next steps, it is convenient to explain the relationship between the strain and stress vectors. According to Lamé’s equations [24], the strain vector is related to the stress vector through the compliance matrix as shown in the following equation:
where
(4)
E is the Young’s modulus and ν is the Poisson’s coefficient. Henceforth, it is possible to consider to be a generic zero matrix of -rows and -columns, and to be a generic identity matrix of -order.
2. THE HYDROSTATIC AND DEVIATORIC COMPONENTS OF THE STRESS Any complex stress tensor two separate tensors [9]:
[25], and consequently the vector
, can be expressed as a sum of
(5)
is the hydrostatic stress tensor, which can also be referred to as the mean normal, spherical or volumetric stress tensor: with
is the hydrostatic scalar stress value and hydrostatic stress with the stress state.
where
(6)
is the hydrostatic matrix which correlates
is the deviatoric component (also called stress deviator tensor), which can be expressed as follows:
where
(7)
is the deviatoric matrix which correlates deviatoric stress with the stress state. Vector is made up of nine components that are generically called . It is important for the following consideration remember that first invariant of the deviatoric tensor is equal to zero: . Then corresponding strain deviatoric vector can be expressed by: (8)
It is therefore possible to affirm that the deviatoric stress and deviatoric strain components are proportional. A similar proportional relationship exists between the hydrostatic stress and strain vectors: (9)
In both cases is the same compliance matrix. In general, total stress components resulting in are not proportional to the corresponding total strain components. This result is very relevant to considerations that follow.
3. DEVIATORIC ENERGY DENSITY At this point, it is possible to evaluate the deviatoric energy density
, [26] expressed as: (10)
where the apex identifies the transpose and conjugate operation. With some simple elaborations, using (8) and (7), the previous equation becomes: (11)
Deviatoric energy density is a scalar quantity; it is obtained by the scalar product , according to (10). The strain deviatoric vector is proportional to the corresponding deviatoric stress vector by (8), and consequently each term of the summation (11) is proportional to , which is a scalar product equating to the squared module of the , or, in other terms, the related power spectrum at frequency . Hence, it is a real positive number. For this reason, the deviatoric energy density is a real quantity. This result is not true for total energy density because its terms are in general not real since total stress components are not proportional to the corresponding total strain components.
The nine stress components of the deviatoric stress vector are not linearly independent. Therefore, the relation (7) can be expressed by a QR decomposition of the square matrix . The decomposition is expressed by: (12)
where is an orthogonal matrix and is an upper triangular matrix. It should be recalled that “if has linearly independent columns, then the first columns of form an orthonormal basis for the column space of . More generally, the first columns of form an orthonormal basis for the span of the first columns of for any ” [27]. The fact that any column of depends only on the first columns of is responsible for the triangular form of . Ergo, for the sake of this research, it is preferable to only account for the independent rows of matrix . This allows for the definition of a fictitious deviatoric stress vector which is composed of just five linearly independent terms.
where
(13)
The vector can be defined as “reduced deviatoric stress vector”. It has no physical significance in terms of stress state representation. The properties of this vector are: (14)
The deviatoric energy density, expressed through (11), can be consequently defined as follows: (15)
Assume a uniaxial stress state with unitary value ; the corresponding deterministic deviatoric energy density will be a scalar positive value expressed by: (16)
4. UNIAXIAL EQUIVALENT STRESS EVALUATION In the previous chapters, the authors proposed a detailed mathematical description of all the parameters needed to reach the definition of the uniaxial equivalent stress state. In order to do that, the deviatoric energy density (15) is divided by the scalar value (16). Thus, can be defined as: (17)
This is a square value of the stress module, which is a power spectrum obtained by the summation of the auto-spectra values of each component of the complex vector . In order to simplify the equations, a new upper triangular matrix was defined as a scaled value of :
(18)
Exploiting the (13), the deviatoric reduced vector can be expressed by: (19)
So, the (17) becomes: (20)
If an equivalent uniaxial complex stress state becomes:
is considered the G-equivalent deviatoric ratio
(21)
This represents a generalized form of the von Mises criterion [21], formulated in the complex domain of complex spectra of stress components in each frequency. Furthermore, this formula satisfies the special case of real static stress state. In this case, it gives the same result as the classical von Mises criterion. Then, for a simple case where the function that describes the variation of each stress component is sinusoidal at a particular frequency and zero mean value, it is possible to reduce a multiaxial stress state to a uniaxial equivalent one, , when using a valid energetic approach defined in the complex domain. In this case, the exact value of is determined by the knowledge of the complex spectral vector In general, however, in real cases, even if the matrix is known, the entity of the complex vector remains unknown. To overcome this problem, it is possible to introduce the Cross Spectral Matrix defined as a matrix: (22)
The trace of gives us the value of equation (21). Then, from , it is possible to determine the scalar value . An evaluation of is always possible, also in the case of a complex load history: it can be done from the direct knowledge of auto-spectra and cross-spectra at frequency contained in the Cross Spectral Matrix , or from an estimate of these via an ensemble averaging of the Fourier transform of the time signal brought back to a set of T length samples. This method is an extremely fast and simple way to estimate correctly matrix, even in general time varying stress states. At this point it is important to consider that, for deterministic
sinusoidal cases, the matrix has , meaning that stress components are all linearly related. Such time stress components can be viewed as an output of a Linear system (the input of which is ) with a frequency response function that shows a specific amplitude and a phase shift. Hence, according to the approach suggested in this paper, it is not recommended to speak of non-proportional loading when dealing with stress components at frequency ω with some reciprocal phase shift. They are all proportional, although linearly related by complex frequency responses. At the conclusion of this chapter the authors can affirm that the first aim of this work has been demonstrated: “if properly used, energy evaluation is a useful tool to determine the equivalent uniaxial stress state.” At this point, the authors consider appropriate, to complete the discussion with the demonstration that by using energy methodologies, it is possible to also solve problems related to material behaviour and mean stress state. Such analysis is aimed at proving that the energy approach, defined in the complex domain, and particularly the one expressed in this article, is valid. Furthermore, the introduction of a mean value allows the authors to demonstrate that the results stemming from the application of the energetic method are in line with experimental data.
5. HYDROSTATIC DENSITY ENERGY AND THE TOTAL ENERGY As it has previously been done for the deviatoric energy density, it is useful to consider the energy density related to the hydrostatic stress tensor [28]. In this case, the hydrostatic strain components are proportional to the hydrostatic stress (9) and it is possible to express by the scalar complex value . The formulation of the hydrostatic density energy is: (23)
Following the methodology that was previously used, when dividing hydrostatic energy density by the unitary monoaxial deviatoric energy express by (16), it is possible to write: (24)
At the same time, a matrix
can introduce: (25)
Considering the logical steps introduced in the chapter 4 to define G-equivalent hydrostatic ratio by the same formulation of (20):
, it is possible to express the
(26)
will be called G-equivalent hydrostatic ratio. Analogously it is thinkable that its value comes from an equivalent monoaxial complex stress state:
(27)
At this point we would like to underline that the sum of the total energy density :
and
multiplied by
generates
(28)
6. CORRECTION FOR MATERIALS FATIGUE BEHAVIOUR It is established that the traditional formulation by von Mises overestimates the equivalent stress for brittle materials subjected to shear stress [21]. Suppose that the proposed criterion, based on the same logical approach as von Mises’, was affected by the same error. Von Mises Criterion establishes that the ratio between normal stress limit and shear stress limit is equal to . This represents a limitation, as previously defined, because some materials have a shear stress limit which is greater than normal stress limit divided by ; this is a peculiar characteristic of brittle behaviour. Even if the Authors believe that the correction of this assumption makes sense only for academics, because brittle should not be used in real fatigue applications, in this chapter, they will nevertheless demonstrate that, to overcome this problem, it is possible to use an energetical approach. Consider the ratio between the normal stress limit and shear stress limit , defined by: (29)
This quantity varies between (for completely ductile materials) and 1 (for completely brittle materials). To correct the G-equivalent deviatoric ratio , taking into account this material behaviour, an energetic approach shows the following. Consider the ratio: (30)
It is equal to the square root of the ratio between hydrostatic and deviatoric energy. The value of is 0 for pure shear stress, and equals
for pure normal stress. Such value represents the
maximum that is able to achieve, but the effective value of must be limited to be less than . To correct in order to respect the real material behaviour, a corrective factor equal to is introduced when (pure shear stress), and equal to when (normal stress). G-equivalent deviatoric ratio can be express in function of
: (31)
is the G-equivalent deviatoric corrected ratio, the limit of its value is if (shear stress) and for (normal stress). The behaviour of this correction is shown in cases
of a biaxial real stress state in Figure (1), In which and represent the two principal stress components of a generic biaxial stress state and is the normal stress limit for the corresponding material. The figure shows that the correction is maximum in case of pure shear stress and it vanishes in case of normal stress. This correction fits well with the experimental data.
7. INFLUENCE OF MEAN STRESS VALUE Mechanical components are rarely subjected to a pure alternating stress; usually, in fact, a mean stress value is added to the alternating one. Different methods lead to the quantification of this mean stress state. The Authors in this paper will use Gerber’s parabola equation [29] to combine alternating and mean stress. However, the authors consider the use of other methods present in literature equally valid [30]. Irrespective of the chosen method used to arrive to the definition of an alternating equivalent stress state , the identification of an alternating stress and a mean one remains necessary. can be obtained from value through equations (21) and (31): (32)
To obtain constant
, it is necessary to consider the complete complex time vector (1) in which a real can be added: (33)
Applying the concept expressed in (20) to this equation we will obtain: (34)
Analysing the right part of the equation, the first term of the summation is the equivalent deviatoric power spectrum of , the second term is the equivalent deviatoric power spectrum of and the third term is a sinusoidal function with zero mean stress; the latter must be discarded when evaluating mean effects. Hence, the equivalent deviatoric mean stress should be calculated only with the first term. In some references [31] the hydrostatic part is added to the deviatoric part of the mean stress. The mean equivalent stress is consequently calculated as follows: (35)
The hydrostatic part of mean stress is involved in fatigue because in many cases it plays the role of crack nucleation facilitator; in detail, if the hydrostatic mean stress state is positive, the hydrostatic part of the mean stress increases the value of . Another thing that must be considered is that the maximum value of mean equivalent stress must not exceed the yield stress so, the is related with in order to not become greater than this value. and must be combined to evaluate a final and then compared with the tensile fatigue limit . To do this, we used the Gerber parabola [29] shown in figure (2) which has the general equation:
(36)
Analysing the figure, describes the work point and L represents the corresponding limit point. So, it is thinkable to consider a factor that permits to evaluate the distance between failure limit and work point, which is expressed by the following equation: (37)
Taking into account this relation, it is possible to evaluate the final alternating equivalent monoaxial tensile stress by: (38)
8. EXPERIMENTAL VALIDATION In order to verify the approach and to consolidate the general procedure, a database of experimental tests was used for a preliminary verification of the proposed method [32]. Different multiaxial loads were used to cover, as far as possible, most multiaxial conditions. In Tables 1-4 approximately 50 experimental datasets, taken from literature [33-35], are shown and used to demonstrate that the proposed energetic method is a useful tool to predict fatigue life of components subjected to either in-phase or out-of-phase deterministic loads. From the tables, it is possible to highlight that four different materials with different mechanical characteristics were used. In particular, in the left part, each table reports the material reference: the name of the material, its Young’s modulus expressed in MPa, Poisson’s coefficient , and a series of stress values pertaining to static characteristics and material fatigue behaviour, also expressed in MPa. In the central part of the tables each row represents an Experimental Multiaxial Endurance Fatigue. The stress value is expressed in MPa and the phase shift in degree. In the right part, instead, the tables contain the Predicted Data: the final alternative equivalent monoaxial tensile stress in MPa , its relation with tensile fatigue limit and the error margin expressed in percentage. The data reported in Table 1 and Table 4, refer to the paper from S. Lasserre and C. Froustey [33]. Tests were conducted at room temperature, with a staircase method on 10 to 12 specimens; each one these traction-torsion tests were carried out after cycles. Also, in Table2 the tests were performed by the staircase method on 15 specimens of 25CrMo4, but in this case the considered limit is of cycles; the tests, conducted by A. Troost and reported in Table3, instead, were carried out after cycles; data stem from the staircase method on 15 specimens. To facilitate the reader in evaluating the accuracy of the predicted data, the Authors decided to show the results in terms of final divided by in Figure (3). The ratio represents a correctly predicted value, every deviation from it shows an underlying error. All of the predicted failure points remain in a gap of from the real behaviour. As it is possible to verify in Table (4), only the last value of the experimental data exceeds error range previously defined, but remain in a gap of respect to the real value. Considering that the uncertainty in the experimental data can be superior to 8%, the proposed method shows a good accuracy.
9. CONCLUSIONS The present paper has shown that through a correct procedure it is possible to reduce a multiaxial stress state to a uniaxial equivalent one, using an energetical approach formulated in the frequency domain. Thus, it is possible to affirm that the energy quantity in a system is related to the energy contained in stress signals. This relationship satisfies a particular case of static stress where the proposed criterion gives the same results as the classical continuum mechanics formulation of von Mises. If an energetic approach is used, traditional criteria cannot be applied through conventional procedures when a multiaxial stress state is involved, but rather a mathematical and conceptual transformation is required in order to evaluate it correctly. In particular, auto and cross-power spectra are the essential mathematical functions used to indicate the whole damage. For the simple deterministic stress state cases, where the stress varies at unique frequency with mean static components, the method is drastically simpler than other methods proposed in literature. The method object of this work could also be used in the context of non-deterministic multiaxial stress cases, when simultaneously analyzing different frequencies , which make up loads. The comparison that was made between theoretical predictions and experimental data shows the effectiveness of the proposed method, which could help engineers verify the fatigue limit of a structure stressed by multiaxial loads. The method is also very simple to implement; for this reason, it is thinkable to apply it to large structures via FE analysis [36].
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[36] C. Braccesi, F. Cianetti, L. Tomassini, “An innovative modal approach for frequency domain stress recovery and fatigue damage evaluation”, Int. J. of Fatigue,91, 2016, 382-396 pp.
1.5
1
k=1.7321 k=1.5 k=1.2 k=1
0
2
n
s /S [no units]
0.5
-0.5
-1
-1.5 -1.5
-1
-0.5
0 0.5 s /S [no units] 1
Figure 1: failure loci for biaxial real stress state
1
1.5
n
, relating to the variation of the parameter
1
L
a /Sn [no units]
0.8
P 0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
m /Su [no units] Figure 2: Gerber parabola with working point P
and the limit point L
1.1
1
aeq
n
/S [no units]
1.05
0.95
0.9 30NCD16 0.85
0
5
25CrMo4 10
15
34Cr4
20 25 30 35 Experimental Data ID
Figure 3: Ratio between experimental results (
40
XC18 45
50
for Data sets ID no. 14) and the estimated
Table 1: Material characteristics, experimental conditions and predicted data for data ID no.1-10 (30NCD16 ref. [33]) Reference
Material
[33]
30NCD16
[MPa] 195000
0.29
[MPa]
[MPa]
[MPa]
[MPa]
1160
1020
710
450
[°] 0 0 0 90 0 0 0 45 60 90
[MPa] 710.00 710.00 667.98 660.97 710.23 706.68 727.36 727.36 719.42 721.89
Test No. 1 2 3 4 5 6 7 8 9 10
[MPa] 0 0 0 0 300 300 300 300 300 300
[MPa] 710 0 485 480 630 0 480 480 470 473
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 450 280 277 0 395 277 277 271 273
[°] 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0
1.00 1.00 0.94 0.93 1.00 1.00 1.02 1.02 1.01 1.02
[%] 0.00 0.00 5.92 6.91 -0.03 0.47 -2.45 -2.45 -1.33 -1.68
Table 2: Material characteristics, experimental conditions and predicted data for data ID no. 11-29 (25CrMo4 ref. [34]) Reference
Material
[34]
25CrMo4
[MPa] 210000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
520
350
310
179
Test No. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
[MPa] 0 0 -170 150 -338 450 340 170 170 170 170 0 0 0 340 340 340 340 470
[MPa] 361 0 362 336 338 261 289 261 275 240 196 270 261 277 220 233 155 159 235
[MPa] 0 0 0 0 0 0 170 340 340 340 340 0 0 0 170 170 170 170 235
[MPa] 0 0 0 0 0 0 144,5 261 275 240 196 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 228 0 0 0 0 0 0 0 0 0 135 131 139 110 117 155 159 118
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 60 90 180 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0 0 0 60 90 60 90 60 90 0
[MPa] 361,00 361,00 369,45 359,81 367,59 356,62 352,34 356,62 393,64 395,13 379,37 349,69 338,55 359,28 367,03 375,32 372,58 376,34 376,61
1,00 1,00 1,02 1,00 1,02 0,99 0,98 0,99 1,09 1,09 1,05 0,97 0,94 1,00 1,02 1,04 1,03 1,04 1,04
[%] 0,00 0,00 -2,34 0,33 -1,82 1,21 2,40 1,21 -9,04 -9,45 -5,09 3,13 6,22 0,48 -1,67 -3,97 -3,21 -4,25 -4,32
Table 3: Material characteristics, experimental conditions and predicted data for data ID no. 30-48 (34Cr4 ref. [35]) Reference
Material
[35]
34Cr4
[MPa] 220000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
780
660
361
228
Test No. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 279 284 0 212 0
[MPa] 410 0 314 218 122 382 315 316 315 224 380 316 314 315 279 284 355 212 129
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 158 157 157,5 0 0 177,5 0 0
[MPa] 0 256 157 218 244 95 157,5 158 157,5 224 95 158 157 157,5 140 142 88,8 212 258
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 60 90 120 90 90 0 60 90 0 90 0 90 90
[MPa] 410,00 410,00 407,75 419,58 415,93 413,38 409,04 410,34 409,04 431,12 411,53 446,58 444,89 445,73 417,90 421,24 430,38 445,07 439,79
1,00 1,00 0,99 1,02 1,01 1,01 1,00 1,00 1,00 1,05 1,00 1,09 1,09 1,09 1,02 1,03 1,05 1,09 1,07
[%] 0,00 0,00 0,55 -2,34 -1,45 -0,82 0,23 -0,08 0,23 -5,15 -0,37 -8,92 -8,51 -8,71 -1,93 -2,74 -4,97 -8,55 -7,27
Table 4: Material characteristics, experimental conditions and predicted data for data ID no. 49-53 (XC18 ref. [33]) Reference
Material
[33]
XC18
[MPa] 218000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
795
657
410
256
Test No. 49 50 51 52 53
[MPa] 0 0 0 0 0
[MPa] 310 0 230 230 242
[MPa] 0 0 0 0 0
[MPa] 0 0 0 0 0
[MPa] 0 0 0 0 0
[MPa] 0 179 133 133 140
[°] 0 0 0 0 0
[°] 0 0 0 0 0
[°] 0 0 0 45 90
[MPa] 310,00 310,00 325,51 325,51 342,57
1,00 1,00 1,05 1,05 1,11
[%] 0,00 0,00 -5,00 -5,00 -10,51
Highlights
Simple Damage Energy Criterium for life prediction in multiaxial fatigue. Through frequency domain a multiaxial stress state can be reduced. Auto and cross power spectra are the fundamental mathematical functions to do this. Comparison between theoretical predictions and experimental data.
S0142-1123(18)30087-2 https://doi.org/10.1016/j.ijfatigue.2018.03.003 JIJF 4604
To appear in:
International Journal of Fatigue
Received Date: Revised Date: Accepted Date:
27 November 2017 2 March 2018 3 March 2018
Please cite this article as: Braccesi, C., Morettini, G., Cianetti, F., Palmieri, M., Development of a new simple energy method for life prediction in multiaxial fatigue, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/ j.ijfatigue.2018.03.003
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DEVELOPMENT OF A NEW SIMPLE ENERGY METHOD FOR LIFE PREDICTION IN MULTIAXIAL FATIGUE C. BRACCESI, G. MORETTINI(o), F. CIANETTI, M. PALMIERI University of Perugia, Department of Industrial Engineering, Via. G. Duranti 1/A-4 - 06125 Perugia – Italy (o) corresponding author, e-mail: [email protected], tel: +39 329 0049 720
Abstract. In this paper, a new, simple Damage Energy Criterium for life prediction in multiaxial fatigue, developed in the frequency domain, is described and verified. The first aim of this paper is to demonstrate that through a correct use of the energetic approach, developed in the frequency domain, it is possible to achieve an accurate result. The proposed theoretical procedure finds its conceptual foundation in the fact that the energy criterion cannot be directly applied to usual stress state in the time domain; however, with a simple passage it is possible to accurately reduce a multiaxial stress state to an equivalent uniaxial one. For this reason, this work will be characterized by analytical content and by substantial mathematical demonstrations. At the end, in order to verify the method and consolidate the general procedure, some experimental tests are used to corroborate the validity of this criterion. This proposal shows drastically simpler technique than any other proposed in current literature.
Keywords: multiaxial fatigue method; fatigue life prediction; energy multiaxial method; damage criteria; energy density criteria.
NOMENCLATURE Zeros matrix of -rows and -columns Compliance matrix Deviatoric matrix operator E
Young’s modulus Complex strain vector Deviatoric complex strain tensor Hydrostatic complex strain vector Cross Spectral Matrix of Cross Spectral Matrix of Hydrostatic matrix operator Identity matrix of -order Imaginary unit Fatigue limits ratio
ν
Poisson’s coefficient
Orthogonal matrix of QR decomposition of Upper deviatoric triangular matrix, QR decomposition of “real part” operator Upper hydrostatic triangular matrix Complex stress tensor Complex stress vector Deviatoric complex stress tensor Hydrostatic complex stress tensor Complex spectrum of alternating stress vector Deviatoric complex stress components Hydrostatic complex stress scalar value Complex stress components Complex spectrum of mean stress vector Tensile stress fatigue limit Ultimate tensile strength Yield tensile strength Alternating stress Eequivalent alternating sigma Stress components in time domain Mean stress t
Time Torsion stress fatigue limit Total energy density Deviatoric energy density Hydrostatic energy density
ω
Frequency
INTRODUCTION Multiaxial fatigue is a topic of great interest for industry and academic world. Therefore, the fatigue design of structural components is actually subject to considerable attention. In the scientific community, there is no universally accepted criteria regarding the methodology for the study of such a problem, although it is worth mentioning that in recent years, several methods have been presented in the literature. It is possible to classify these methods into different categories depending on the parameters and the approach with which the phenomenon is analysed. The first significant distinction stems from dividing the criteria into two categories: Safe/Unsafe Criteria and Damage Criteria. The first category determines whether the component, subjected to a multiaxial stress state, is resistant or not. To better analyse these criteria, it is useful to refer to their graphic interpretation. Indeed, each of these [1-3] defines one or more failure straight lines that split the plain, on which the loading path can be drawn, in a safety and unsafety domain [4]. A loading path that remains into bounding failure lines is expected to have infinite life while any path that extends outside the damage line will have fatigue failures. With this type of approach, only a safety factor for fatigue limit can be evaluated. F. Morel [5] was the first who tried to apply this approach to the damage evaluation concept. With his work, he tried to formulate a method to evaluate the damage. Through the second category, (Damage Criteria), it is possible to evaluate the damage caused by load time history as well as the damage contribution of a part of the stress signal. To this day, most multiaxial fatigue criteria fall under this category. Within the Damage Criteria it is possible to make a further distinction based on the approach with which the damage is calculated: the Geometric Approach and the Energetic Approach. The Geometric approach is also known as “Critical Plane” approach, which has been evolving since W. N. Findley [6] first proposed his critical plane-based criterion in 1959. This approach is based on the experimental observation that fatigue cracks initiate and grow on a certain material plane. Only stress and/or strain components acting on the critical plane are responsible for fatigue failure of the material. The chosen geometric reference system, and then of the critical plane, can happen either a priori or by iteration, using the maximization of some stress and/or strain parameters. According to Matake [7] the critical plane orientation equals the maximum shear stress amplitude, whereas according to Z. Q. Tao [8], it equals the maximum shear strain. For the other [9-14] this particular parameter could vary. The Critical Plane Damage criteria require calculation times which depend on the number of physical planes investigated during the fatigue limit estimation. It is clear that the fatigue limit is correlated to the micro-structure of the material [15] and thus to its geometry; however, the Energy Damage criteria are based on the assumption that it is possible to analyse the damage fatigue phenomenon using another parameter, the energy density, which is independent on the geometric reference system of choice. Energy density thus represents the only parameter necessary for the estimation of fatigue limit of metallic structural components subject to a multiaxial stress state. The first researcher to use an energy-based criterion for multiaxial fatigue was Y. S. Garud [16]. In 1981, he proposed an energy criterion associated with plastic deformation. Subsequently, F. Ellyin [17] proposed an approach based on the combination of elastic and plastic strain energy which demonstrates that these quantities, [18], result from the slipping of more than one plane. In the last decades, different Damage Energy criteria have been proposed in literature, such as the Park criterion or Susmel criterion [19-20]. The use of the Energy criteria presents a fundamental advantage when compared with the criteria discussed above: the energy quantity is independent on the component’s geometry. Through the use of the energy value alone, it is possible to evaluate the damage caused by load time history very quickly. As for the other categories of methods, Damage Energy Criteria can be classified into two categories that depend on the domain these methodologies have been developed in: the Time Domain and the Frequency Domain. Most of the times, loading conditions of a system are expressed in the Time Domain; these signals can thus be used without any pre-processing. The great advantage of this methodology is that there
are no limitations in terms of statistical distribution of the cycle. On the other hand, however, time stress histories are usually random, so in the time domain, even if it is possible to find the complete load history, it will be impossible it will be impossible to find load cycles and consequently an average energy value [17]. The energy expressed in this domain is a squared value and carries errors related to the doubling of the stress signal frequency. The authors affirm that this is the fundamental reason why the use of energy methods have come to be underestimated by the scientific community. The authors will show that solution to the above problems can be found in a procedure involving the use of Frequency Domain analysis. This subclass of methods holds that a simple elaboration of the signal is sufficient to correctly evaluate the energy of a given system. With this work, the authors propose a new simple Damage Energy Criterion for life prediction in multiaxial fatigue, developed in the Frequency Domain. The first aim of this paper is to demonstrate that through the correct use of the energetic approach, developed in the frequency domain, it is possible to achieve an accurate result. For this reason, this work will be characterized by both analytical content and substantial mathematical demonstrations. In particular, the first section of the paper introduces the stress tensor, its definition in the frequency domain and its relationship with the strain tensor. In the second section, the analysis focuses on the utility of dividing the stress tensor in two parts: the deviatoric and the hydrostatic one. In the third section it is then possible express the deviatoric energy density in the frequency domain. At the beginning of the fourth section all of the basic theories are introduced to describe the correct procedure to evaluate the equivalent alternating stress by an energetic approach. The fourth section is of paramount importance for this work; in fact, the most significant conclusions are expressed in this section by the authors. In order to complete the examination, the following chapters demonstrate that it is possible to extend the energy processing to eliminate errors stemming from the behavior of fatigue materials, as well as errors stemming from the influence of the mean stress. Such procedure is based on a precedent work [4] and presents a formal and substantial analogy with the von Mises theory [21]; indeed, for the special case of real static stress state, it gives the same result as the classical von Mises criterion. At the end of the paper, experimental validation is presented to confirm that the technique is drastically simpler than others proposed in literature but has a good accuracy. The simplicity of this method could assist engineers in verifying the fatigue limit of a structure stressed by multiaxial loads, especially during the initial design phase.
1. STRESS TENSOR, ITS DEFINITON AND ITS RELATIONSHIP WITH THE STRAIN TENSOR Consider a state of stress in a point described by a time varying stress tensor. Assume that the function that describes the variation of each stress component is sinusoidal with a particular frequency and a mean value of zero. By Euler’s formula [22], which establishes the fundamental relationship between trigonometric and complex exponential functions, it is possible to express each stress component of the time signal as real part of a complex quantity described by the following relation: (1)
where is the real part and is the imaginary part of the stress; these values can be considered as Fourier’s coefficients according to Fourier analysis of a time signal [23]. At this point, it is useful to focus on the complex representation of each stress component and generate the complex stress tensor (2), where each element is the complex spectrum of the corresponding deterministic time signals. This tensor represents the complex spectra for a particular frequency ω and it characterizes
the stress in a specific component point. Complex tensor components can be arranged in a complex spectral vector , as shown below: (2)
The symbol ' refers to the transpose operation. For the sake of simplicity, it could be useful to reduce, the complex stress vector in another one that only contains six complex components: (3)
In order to complete our examination, and to better analyse the next steps, it is convenient to explain the relationship between the strain and stress vectors. According to Lamé’s equations [24], the strain vector is related to the stress vector through the compliance matrix as shown in the following equation:
where
(4)
E is the Young’s modulus and ν is the Poisson’s coefficient. Henceforth, it is possible to consider to be a generic zero matrix of -rows and -columns, and to be a generic identity matrix of -order.
2. THE HYDROSTATIC AND DEVIATORIC COMPONENTS OF THE STRESS Any complex stress tensor two separate tensors [9]:
[25], and consequently the vector
, can be expressed as a sum of
(5)
is the hydrostatic stress tensor, which can also be referred to as the mean normal, spherical or volumetric stress tensor: with
is the hydrostatic scalar stress value and hydrostatic stress with the stress state.
where
(6)
is the hydrostatic matrix which correlates
is the deviatoric component (also called stress deviator tensor), which can be expressed as follows:
where
(7)
is the deviatoric matrix which correlates deviatoric stress with the stress state. Vector is made up of nine components that are generically called . It is important for the following consideration remember that first invariant of the deviatoric tensor is equal to zero: . Then corresponding strain deviatoric vector can be expressed by: (8)
It is therefore possible to affirm that the deviatoric stress and deviatoric strain components are proportional. A similar proportional relationship exists between the hydrostatic stress and strain vectors: (9)
In both cases is the same compliance matrix. In general, total stress components resulting in are not proportional to the corresponding total strain components. This result is very relevant to considerations that follow.
3. DEVIATORIC ENERGY DENSITY At this point, it is possible to evaluate the deviatoric energy density
, [26] expressed as: (10)
where the apex identifies the transpose and conjugate operation. With some simple elaborations, using (8) and (7), the previous equation becomes: (11)
Deviatoric energy density is a scalar quantity; it is obtained by the scalar product , according to (10). The strain deviatoric vector is proportional to the corresponding deviatoric stress vector by (8), and consequently each term of the summation (11) is proportional to , which is a scalar product equating to the squared module of the , or, in other terms, the related power spectrum at frequency . Hence, it is a real positive number. For this reason, the deviatoric energy density is a real quantity. This result is not true for total energy density because its terms are in general not real since total stress components are not proportional to the corresponding total strain components.
The nine stress components of the deviatoric stress vector are not linearly independent. Therefore, the relation (7) can be expressed by a QR decomposition of the square matrix . The decomposition is expressed by: (12)
where is an orthogonal matrix and is an upper triangular matrix. It should be recalled that “if has linearly independent columns, then the first columns of form an orthonormal basis for the column space of . More generally, the first columns of form an orthonormal basis for the span of the first columns of for any ” [27]. The fact that any column of depends only on the first columns of is responsible for the triangular form of . Ergo, for the sake of this research, it is preferable to only account for the independent rows of matrix . This allows for the definition of a fictitious deviatoric stress vector which is composed of just five linearly independent terms.
where
(13)
The vector can be defined as “reduced deviatoric stress vector”. It has no physical significance in terms of stress state representation. The properties of this vector are: (14)
The deviatoric energy density, expressed through (11), can be consequently defined as follows: (15)
Assume a uniaxial stress state with unitary value ; the corresponding deterministic deviatoric energy density will be a scalar positive value expressed by: (16)
4. UNIAXIAL EQUIVALENT STRESS EVALUATION In the previous chapters, the authors proposed a detailed mathematical description of all the parameters needed to reach the definition of the uniaxial equivalent stress state. In order to do that, the deviatoric energy density (15) is divided by the scalar value (16). Thus, can be defined as: (17)
This is a square value of the stress module, which is a power spectrum obtained by the summation of the auto-spectra values of each component of the complex vector . In order to simplify the equations, a new upper triangular matrix was defined as a scaled value of :
(18)
Exploiting the (13), the deviatoric reduced vector can be expressed by: (19)
So, the (17) becomes: (20)
If an equivalent uniaxial complex stress state becomes:
is considered the G-equivalent deviatoric ratio
(21)
This represents a generalized form of the von Mises criterion [21], formulated in the complex domain of complex spectra of stress components in each frequency. Furthermore, this formula satisfies the special case of real static stress state. In this case, it gives the same result as the classical von Mises criterion. Then, for a simple case where the function that describes the variation of each stress component is sinusoidal at a particular frequency and zero mean value, it is possible to reduce a multiaxial stress state to a uniaxial equivalent one, , when using a valid energetic approach defined in the complex domain. In this case, the exact value of is determined by the knowledge of the complex spectral vector In general, however, in real cases, even if the matrix is known, the entity of the complex vector remains unknown. To overcome this problem, it is possible to introduce the Cross Spectral Matrix defined as a matrix: (22)
The trace of gives us the value of equation (21). Then, from , it is possible to determine the scalar value . An evaluation of is always possible, also in the case of a complex load history: it can be done from the direct knowledge of auto-spectra and cross-spectra at frequency contained in the Cross Spectral Matrix , or from an estimate of these via an ensemble averaging of the Fourier transform of the time signal brought back to a set of T length samples. This method is an extremely fast and simple way to estimate correctly matrix, even in general time varying stress states. At this point it is important to consider that, for deterministic
sinusoidal cases, the matrix has , meaning that stress components are all linearly related. Such time stress components can be viewed as an output of a Linear system (the input of which is ) with a frequency response function that shows a specific amplitude and a phase shift. Hence, according to the approach suggested in this paper, it is not recommended to speak of non-proportional loading when dealing with stress components at frequency ω with some reciprocal phase shift. They are all proportional, although linearly related by complex frequency responses. At the conclusion of this chapter the authors can affirm that the first aim of this work has been demonstrated: “if properly used, energy evaluation is a useful tool to determine the equivalent uniaxial stress state.” At this point, the authors consider appropriate, to complete the discussion with the demonstration that by using energy methodologies, it is possible to also solve problems related to material behaviour and mean stress state. Such analysis is aimed at proving that the energy approach, defined in the complex domain, and particularly the one expressed in this article, is valid. Furthermore, the introduction of a mean value allows the authors to demonstrate that the results stemming from the application of the energetic method are in line with experimental data.
5. HYDROSTATIC DENSITY ENERGY AND THE TOTAL ENERGY As it has previously been done for the deviatoric energy density, it is useful to consider the energy density related to the hydrostatic stress tensor [28]. In this case, the hydrostatic strain components are proportional to the hydrostatic stress (9) and it is possible to express by the scalar complex value . The formulation of the hydrostatic density energy is: (23)
Following the methodology that was previously used, when dividing hydrostatic energy density by the unitary monoaxial deviatoric energy express by (16), it is possible to write: (24)
At the same time, a matrix
can introduce: (25)
Considering the logical steps introduced in the chapter 4 to define G-equivalent hydrostatic ratio by the same formulation of (20):
, it is possible to express the
(26)
will be called G-equivalent hydrostatic ratio. Analogously it is thinkable that its value comes from an equivalent monoaxial complex stress state:
(27)
At this point we would like to underline that the sum of the total energy density :
and
multiplied by
generates
(28)
6. CORRECTION FOR MATERIALS FATIGUE BEHAVIOUR It is established that the traditional formulation by von Mises overestimates the equivalent stress for brittle materials subjected to shear stress [21]. Suppose that the proposed criterion, based on the same logical approach as von Mises’, was affected by the same error. Von Mises Criterion establishes that the ratio between normal stress limit and shear stress limit is equal to . This represents a limitation, as previously defined, because some materials have a shear stress limit which is greater than normal stress limit divided by ; this is a peculiar characteristic of brittle behaviour. Even if the Authors believe that the correction of this assumption makes sense only for academics, because brittle should not be used in real fatigue applications, in this chapter, they will nevertheless demonstrate that, to overcome this problem, it is possible to use an energetical approach. Consider the ratio between the normal stress limit and shear stress limit , defined by: (29)
This quantity varies between (for completely ductile materials) and 1 (for completely brittle materials). To correct the G-equivalent deviatoric ratio , taking into account this material behaviour, an energetic approach shows the following. Consider the ratio: (30)
It is equal to the square root of the ratio between hydrostatic and deviatoric energy. The value of is 0 for pure shear stress, and equals
for pure normal stress. Such value represents the
maximum that is able to achieve, but the effective value of must be limited to be less than . To correct in order to respect the real material behaviour, a corrective factor equal to is introduced when (pure shear stress), and equal to when (normal stress). G-equivalent deviatoric ratio can be express in function of
: (31)
is the G-equivalent deviatoric corrected ratio, the limit of its value is if (shear stress) and for (normal stress). The behaviour of this correction is shown in cases
of a biaxial real stress state in Figure (1), In which and represent the two principal stress components of a generic biaxial stress state and is the normal stress limit for the corresponding material. The figure shows that the correction is maximum in case of pure shear stress and it vanishes in case of normal stress. This correction fits well with the experimental data.
7. INFLUENCE OF MEAN STRESS VALUE Mechanical components are rarely subjected to a pure alternating stress; usually, in fact, a mean stress value is added to the alternating one. Different methods lead to the quantification of this mean stress state. The Authors in this paper will use Gerber’s parabola equation [29] to combine alternating and mean stress. However, the authors consider the use of other methods present in literature equally valid [30]. Irrespective of the chosen method used to arrive to the definition of an alternating equivalent stress state , the identification of an alternating stress and a mean one remains necessary. can be obtained from value through equations (21) and (31): (32)
To obtain constant
, it is necessary to consider the complete complex time vector (1) in which a real can be added: (33)
Applying the concept expressed in (20) to this equation we will obtain: (34)
Analysing the right part of the equation, the first term of the summation is the equivalent deviatoric power spectrum of , the second term is the equivalent deviatoric power spectrum of and the third term is a sinusoidal function with zero mean stress; the latter must be discarded when evaluating mean effects. Hence, the equivalent deviatoric mean stress should be calculated only with the first term. In some references [31] the hydrostatic part is added to the deviatoric part of the mean stress. The mean equivalent stress is consequently calculated as follows: (35)
The hydrostatic part of mean stress is involved in fatigue because in many cases it plays the role of crack nucleation facilitator; in detail, if the hydrostatic mean stress state is positive, the hydrostatic part of the mean stress increases the value of . Another thing that must be considered is that the maximum value of mean equivalent stress must not exceed the yield stress so, the is related with in order to not become greater than this value. and must be combined to evaluate a final and then compared with the tensile fatigue limit . To do this, we used the Gerber parabola [29] shown in figure (2) which has the general equation:
(36)
Analysing the figure, describes the work point and L represents the corresponding limit point. So, it is thinkable to consider a factor that permits to evaluate the distance between failure limit and work point, which is expressed by the following equation: (37)
Taking into account this relation, it is possible to evaluate the final alternating equivalent monoaxial tensile stress by: (38)
8. EXPERIMENTAL VALIDATION In order to verify the approach and to consolidate the general procedure, a database of experimental tests was used for a preliminary verification of the proposed method [32]. Different multiaxial loads were used to cover, as far as possible, most multiaxial conditions. In Tables 1-4 approximately 50 experimental datasets, taken from literature [33-35], are shown and used to demonstrate that the proposed energetic method is a useful tool to predict fatigue life of components subjected to either in-phase or out-of-phase deterministic loads. From the tables, it is possible to highlight that four different materials with different mechanical characteristics were used. In particular, in the left part, each table reports the material reference: the name of the material, its Young’s modulus expressed in MPa, Poisson’s coefficient , and a series of stress values pertaining to static characteristics and material fatigue behaviour, also expressed in MPa. In the central part of the tables each row represents an Experimental Multiaxial Endurance Fatigue. The stress value is expressed in MPa and the phase shift in degree. In the right part, instead, the tables contain the Predicted Data: the final alternative equivalent monoaxial tensile stress in MPa , its relation with tensile fatigue limit and the error margin expressed in percentage. The data reported in Table 1 and Table 4, refer to the paper from S. Lasserre and C. Froustey [33]. Tests were conducted at room temperature, with a staircase method on 10 to 12 specimens; each one these traction-torsion tests were carried out after cycles. Also, in Table2 the tests were performed by the staircase method on 15 specimens of 25CrMo4, but in this case the considered limit is of cycles; the tests, conducted by A. Troost and reported in Table3, instead, were carried out after cycles; data stem from the staircase method on 15 specimens. To facilitate the reader in evaluating the accuracy of the predicted data, the Authors decided to show the results in terms of final divided by in Figure (3). The ratio represents a correctly predicted value, every deviation from it shows an underlying error. All of the predicted failure points remain in a gap of from the real behaviour. As it is possible to verify in Table (4), only the last value of the experimental data exceeds error range previously defined, but remain in a gap of respect to the real value. Considering that the uncertainty in the experimental data can be superior to 8%, the proposed method shows a good accuracy.
9. CONCLUSIONS The present paper has shown that through a correct procedure it is possible to reduce a multiaxial stress state to a uniaxial equivalent one, using an energetical approach formulated in the frequency domain. Thus, it is possible to affirm that the energy quantity in a system is related to the energy contained in stress signals. This relationship satisfies a particular case of static stress where the proposed criterion gives the same results as the classical continuum mechanics formulation of von Mises. If an energetic approach is used, traditional criteria cannot be applied through conventional procedures when a multiaxial stress state is involved, but rather a mathematical and conceptual transformation is required in order to evaluate it correctly. In particular, auto and cross-power spectra are the essential mathematical functions used to indicate the whole damage. For the simple deterministic stress state cases, where the stress varies at unique frequency with mean static components, the method is drastically simpler than other methods proposed in literature. The method object of this work could also be used in the context of non-deterministic multiaxial stress cases, when simultaneously analyzing different frequencies , which make up loads. The comparison that was made between theoretical predictions and experimental data shows the effectiveness of the proposed method, which could help engineers verify the fatigue limit of a structure stressed by multiaxial loads. The method is also very simple to implement; for this reason, it is thinkable to apply it to large structures via FE analysis [36].
REFERENCES [1] K. Dang Van, I.V. Papadopoulos. “Multiaxial fatigue failure criterion: a new approach”. Third Int. Conf. on Fatigue Thresholds-Fatigue, EMAS Warley U. K, 1987, 997-1008 pp. [2] I. V. Papadopoulos, P. Davoli, C. Gorla, M. Filippini, A. Bernasconi, “A comparative study of multiaxial high-cycle fatigue criteria for metals.”, Int. J. of Fatigue 19, 1996, 219-235 pp. [3] B. Crossland, “Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel.”, Proceedings of International Conference on Fatigue of Metals, Institution of Mechanical Engineering, London, 1956, 138-149 pp. [4] C. Braccesi, F. Cianetti, L. Guido, D. Pioli, “An equivalent stress process for fatigue life estimation of mechanical components under multiaxial stress condition”, Int. J. of Fatigue, 2008, 1479-1497 pp. [5] F. Morel, “A critical plane approach for life prediction of high cycle fatigue under multiaxial variable amplitude loading”, Int. J. Fatigue, 2000, 101-119 pp. [6] W.N. Findley, “A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending.”, J. of Eng. for Ind., ASME 81, 1959, 301-306 pp. [7] T. Matake, “An explanation on fatigue limit under combined stress.”, Bulletin of the Japan Society of Mechanical Engineers 20, 1977, 257-263 pp. [8] Z. Tao, D. Shang, H. Liu, H. Chen, “Life prediction based on weight-averaged maximum shear strain range plane under multiaxial variable amplitude loading”, Fatigue and Fracture of Engineering Materials and Structures, 2016, 907-920 pp. [9] D.L. McDiarmid, “A general criterion for high cycle multiaxial fatigue failure.”, Fatigue & Fracture of Engineering Materials and Structures 14, 1991, 429-453 pp. [10] A. Karolczuk, “Plastic strains and the macroscopic critical plane orientations under combined bending and torsion with constant and variable amplitudes.”, Engineering Fracture Mechanics 73, 2006, 1629-1652 pp. [11] A. Carpinteri, R.Brighenti, A. Spagnoli, “A fracture plane approach in multiaxial highcycle fatigue of metals.”, Fatigue & Fracture of Engineering Materials and Structures 23, 2000, 355-364 pp.
[12] T.Łagoda, P.Ogonowski, “Criteria of multiaxial random fatigue based on stress, strain and energy parameters of damage in the critical plane.”, Materialwissenschaft und Werkstofftechnik 36, 2005, 429-437 pp. [13] C. Wang, D.G. Shang, X.W. Wang, “A new multiaxial high-cycle fatigue criterion based on the critical plane for ductile and brittle materials.”, Journal of Materials Engineering and Performance 24, 2015, 816-824 pp [14] J. Araújo, A. Dantas, F. Castro, E. Mamiya, J. Ferreira, “On the characterization of the critical plane with a simple and fast alternative measure of the shear stress amplitude in multiaxial fatigue”, Int. J. of Fatigue, 2011, 1092-1100 pp. [15] A. Di Schino, M Barteri, J. M. Kenny, “Fatigue behavior of a high nitrogen austenitic stainless steel as a function of its grain size”, journal of materials science letters, 2003, 1511-1513 pp. [16] Y.S. Garud, “A new approach to the evaluation of fatigue under multiaxial loadings.”, ASME Journal of Engineering Materials and Technology, 1981, 118-125 pp [17] F. Ellyin, “Multiaxial fatigue-a perspective”, Key Engineering Materials, 2007, 205-210 pp. [18] F. Ellyin, “Fatigue Damage, Crack Growth and Life Prediction”, Springer Science & Business Media, 2012, 156-172 pp. [19] J. Park, D. Nelson, “ Evaluation of an energy-based approach and a critical plane approach for predicting constant amplitude multiaxial fatigue life.”, Int. J. of Fatigue 22, 2000, 23-39 pp. [20] L. Susmel, P. Lazzarin, “A stress-based method to predict lifetime under multiaxial fatigue loadings.”, Fatigue & Fracture of Engineering Materials and Structures 26, 2013, 1171-1187 pp. [21] R. Von Mises, "Mechanik der Festen Korper im Plastisch Deformablen Zustand," Nachr. Ges. Wiss. Gottingen, 1913, 582 pp. [22] L. Euler, “Miscellanea Berolinensia ”, vol.7, 1743, 193–242 pp. [23] P. Duhamel, M. Vetterli, "Fast Fourier transforms: a tutorial review and a state of the art", Signal Processing, 19, 1990, 259–299 pp. [24] N. Kh. Rozov, "Lamé equation", Encyclopedia of Mathematics, Springer, 2001. [25] G. Thomas, E. George, "Continuum Mechanics for Engineers", CRC Press, 1999. [26] C. Rabindranath, "Mathematical Theory of Continuum Mechanics", Alpha Science, 1999. [27] L. N. Trefethen, D. Bau, “Numerical Linear Algebra”, SIAM, 1997. [28] I-Shih Liu, "Continuum Mechanics", Springer, 2002. [29] J. Schijve, “Fatigue of Structures and Materiales”, Springer Science, 2008, 152 pp. [30] Das, P.K., “Effect of mean stress on fatigue properties.”, J. of the Ins. of Engineers, Issue pt ME 4, 1976, 151-155 pp. [31] M. Shimura, S. Saito, “The effects of hydrostatic tensile stress on fracture in some structural alloys”, Science Rep. Research Institutes, Tohoku University, 1983, 201-215 pp. [32] J. Papuga, S. Parma, M. Ruzicka, “Systematic validation of experimental data usable for verifying the multiaxial fatigue prediction methods”, Frattura ed integrità strutturale 38, 2016, 106-113 pp. [33] S.Lasserre , C.Froustey "Multiaxial fatigue of steel-testing out of phase and in blocks: validity and applicability of some criteria", int. J. Fatigue,1992, 113-120 pp. [34] Mielke, "Festigkeitsverhalten metallischer Werkstoffe unter zweiachsiger schwingender Beanspruchung mit verschiedenen Spannungszeitverläufen", [PhD thesis]. 1980 [35] R.Heidenreich, I.Richter, H.Zenner, "Schubspannungsintensitätshpothese–Weitere experimentalle und theorestische Untersuchungen" Konstruktion, Vol 36, 1984, 99-104 pp.
[36] C. Braccesi, F. Cianetti, L. Tomassini, “An innovative modal approach for frequency domain stress recovery and fatigue damage evaluation”, Int. J. of Fatigue,91, 2016, 382-396 pp.
1.5
1
k=1.7321 k=1.5 k=1.2 k=1
0
2
n
s /S [no units]
0.5
-0.5
-1
-1.5 -1.5
-1
-0.5
0 0.5 s /S [no units] 1
Figure 1: failure loci for biaxial real stress state
1
1.5
n
, relating to the variation of the parameter
1
L
a /Sn [no units]
0.8
P 0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
m /Su [no units] Figure 2: Gerber parabola with working point P
and the limit point L
1.1
1
aeq
n
/S [no units]
1.05
0.95
0.9 30NCD16 0.85
0
5
25CrMo4 10
15
34Cr4
20 25 30 35 Experimental Data ID
Figure 3: Ratio between experimental results (
40
XC18 45
50
for Data sets ID no. 14) and the estimated
Table 1: Material characteristics, experimental conditions and predicted data for data ID no.1-10 (30NCD16 ref. [33]) Reference
Material
[33]
30NCD16
[MPa] 195000
0.29
[MPa]
[MPa]
[MPa]
[MPa]
1160
1020
710
450
[°] 0 0 0 90 0 0 0 45 60 90
[MPa] 710.00 710.00 667.98 660.97 710.23 706.68 727.36 727.36 719.42 721.89
Test No. 1 2 3 4 5 6 7 8 9 10
[MPa] 0 0 0 0 300 300 300 300 300 300
[MPa] 710 0 485 480 630 0 480 480 470 473
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0
[MPa] 0 450 280 277 0 395 277 277 271 273
[°] 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0
1.00 1.00 0.94 0.93 1.00 1.00 1.02 1.02 1.01 1.02
[%] 0.00 0.00 5.92 6.91 -0.03 0.47 -2.45 -2.45 -1.33 -1.68
Table 2: Material characteristics, experimental conditions and predicted data for data ID no. 11-29 (25CrMo4 ref. [34]) Reference
Material
[34]
25CrMo4
[MPa] 210000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
520
350
310
179
Test No. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
[MPa] 0 0 -170 150 -338 450 340 170 170 170 170 0 0 0 340 340 340 340 470
[MPa] 361 0 362 336 338 261 289 261 275 240 196 270 261 277 220 233 155 159 235
[MPa] 0 0 0 0 0 0 170 340 340 340 340 0 0 0 170 170 170 170 235
[MPa] 0 0 0 0 0 0 144,5 261 275 240 196 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 228 0 0 0 0 0 0 0 0 0 135 131 139 110 117 155 159 118
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 60 90 180 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0 0 0 60 90 60 90 60 90 0
[MPa] 361,00 361,00 369,45 359,81 367,59 356,62 352,34 356,62 393,64 395,13 379,37 349,69 338,55 359,28 367,03 375,32 372,58 376,34 376,61
1,00 1,00 1,02 1,00 1,02 0,99 0,98 0,99 1,09 1,09 1,05 0,97 0,94 1,00 1,02 1,04 1,03 1,04 1,04
[%] 0,00 0,00 -2,34 0,33 -1,82 1,21 2,40 1,21 -9,04 -9,45 -5,09 3,13 6,22 0,48 -1,67 -3,97 -3,21 -4,25 -4,32
Table 3: Material characteristics, experimental conditions and predicted data for data ID no. 30-48 (34Cr4 ref. [35]) Reference
Material
[35]
34Cr4
[MPa] 220000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
780
660
361
228
Test No. 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 279 284 0 212 0
[MPa] 410 0 314 218 122 382 315 316 315 224 380 316 314 315 279 284 355 212 129
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[MPa] 0 0 0 0 0 0 0 0 0 0 0 158 157 157,5 0 0 177,5 0 0
[MPa] 0 256 157 218 244 95 157,5 158 157,5 224 95 158 157 157,5 140 142 88,8 212 258
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
[°] 0 0 0 0 0 0 60 90 120 90 90 0 60 90 0 90 0 90 90
[MPa] 410,00 410,00 407,75 419,58 415,93 413,38 409,04 410,34 409,04 431,12 411,53 446,58 444,89 445,73 417,90 421,24 430,38 445,07 439,79
1,00 1,00 0,99 1,02 1,01 1,01 1,00 1,00 1,00 1,05 1,00 1,09 1,09 1,09 1,02 1,03 1,05 1,09 1,07
[%] 0,00 0,00 0,55 -2,34 -1,45 -0,82 0,23 -0,08 0,23 -5,15 -0,37 -8,92 -8,51 -8,71 -1,93 -2,74 -4,97 -8,55 -7,27
Table 4: Material characteristics, experimental conditions and predicted data for data ID no. 49-53 (XC18 ref. [33]) Reference
Material
[33]
XC18
[MPa] 218000
0.3
[MPa]
[MPa]
[MPa]
[MPa]
795
657
410
256
Test No. 49 50 51 52 53
[MPa] 0 0 0 0 0
[MPa] 310 0 230 230 242
[MPa] 0 0 0 0 0
[MPa] 0 0 0 0 0
[MPa] 0 0 0 0 0
[MPa] 0 179 133 133 140
[°] 0 0 0 0 0
[°] 0 0 0 0 0
[°] 0 0 0 45 90
[MPa] 310,00 310,00 325,51 325,51 342,57
1,00 1,00 1,05 1,05 1,11
[%] 0,00 0,00 -5,00 -5,00 -10,51
Highlights
Simple Damage Energy Criterium for life prediction in multiaxial fatigue. Through frequency domain a multiaxial stress state can be reduced. Auto and cross power spectra are the fundamental mathematical functions to do this. Comparison between theoretical predictions and experimental data.