The local stress-strain approach to fatigue analysis

TWI Bulletin, January 1985

by Tim Rosenberg

Tim Rosenberg, BSc (Mech Eng), is a Research Engineer in the Fatigue Section of the Engineering Department.

The local stress-strain approach (LSSA) to fatigue analysis aims to predict fatigue crack initiation life from the calculated local stresses and strains at positions of known stress concentration. In the first part of this article, a step by step approach by which the LSSA can be applied is described, and the problems involved with its application to welded joints are discussed.

Part 2 will compare the LSSA approach with the methods normally adopted for analysing welded joints.

In engineering structures, cyclic loading and resulting fatigue make forecasting of service life a necessity. In practice, fatigue cracks often initiate from regions of complex geometry, such as notches and weld toes, under the action of complex load spectra. It is the purpose of the local stress-strain approach (LSSA) to base life predictions on the stresses and strains calculated at these positions, using fatigue data for the material obtained from low cycle fatigue tests on small scale machined specimens.

The main purpose of LSSA is to estimate the number of cycles required to initiate a fatigue crack. Thus, the method has been applied successfully to analysis of unwelded components. For example, it is a well proven technique in the general vehicle and aerospace industries where it has enabled some manufacturers to reduce significantly the time required to develop a new component.

The Welding Institute has long held the view that fatigue crack initiation occupies only a small proportion of the life of a welded joint. This is because welds contain flaws, notably the crack like intrusions which have been observed at weld toes (ref. [2] , chapter 11 ), from which fatigue cracks readily propagate. In view of this, the LSSA seems to be inappropriate for analysis of welded joints. Nevertheless, some research has been carried out into its application to a limited number of welded joint geometries, mainly butt and cruciform. A possible rationale for this lies in the definition of crack initiation: that normally adopted by users of LSSA is production of a crack approximately 0.25mm deep. Weld toe flaws are of the order of 0.15mm deep. Thus, LSSA might be viewed as a method for calculating the number of cycles occupied in the early stages of the life of the fatigue crack.

Alternatively, the method might be suitable for considering those situations in which fatigue cracking does not originate from a crack like flaw, such as higher fatigue strength longitudinal welds or flush ground butt welds containing non-planar defects ( e.g. porosity and slag inclusions).

The LSSA is described in this article and practical aspects of its application to welded joints and its validity are discussed.

Nomenclature

a	material parameter
a I	length of initiated crack
α A	a geometry coefficient
A,B,C	constants in an equation defining K T
b	fatigue strength exponent
c	fatigue ductility exponent
e	nominal strain
Δe	nominal strain range
ε	local strain
Δ ε	local strain range
Δ ε e	elastic strain range
Δ ε p	plastic strain range
	fatigue ductility coefficient
E	Young's Modulus
k	relaxation exponent
	cyclic strength coefficient
K T	monotonic stress concentration factor
K F	fatigue strength reduction factor
K F crit	critical fatigue strength reduction factor
K ε	Neuber strain concentration factor
K σ	Neuber stress concentration factor
	cyclic strain hardening exponent
2N f	reversals to failure (fatigue crack initiation)
2N i	current number of cycles
2N I	fatigue life to initiation
2N t	transitional fatigue life
q	notch sensitivity
r	notch root radius
S	nominal stress
ΔS	nominal stress range
σ	local stress
σa	stress amplitude
Δ σ	local stress range
	fatigue strength coefficient
σ m	mean stress
σ o	current mean stress
σ os	initial mean stress
σ y	yield stress
t	plate thickness
	angles defining the geometry of a butt weld

The local stress-strain approach

Fatigue crack initiation life analysis by the LSSA can be broken down into a number of stages as shown in Fig.1. For example, consider the application of the method to estimating the fatigue life of a member in which the notch at which a fatigue crack will initiate is a weld:

Step 1

Specify the remote stress and strain ranges

Determination of the nominal stress and strain ranges in the loaded member away from the weld is the first requirement. Calculations can be performed for constant or variable amplitude stress-strain spectra. The mean stress can be a significant variable in the analysis and consequently for each stress range in the spectrum the relevant mean stress must be recorded. In the majority of cases there will be no general yielding of the member or component under consideration, but plastic strains may occur at the notch. If this is the case, the nominal and local strain histories will have a qualitative resemblance, but will not be proportional ( Fig.2). Local notch histories can rarely be measured and it is therefore necessary to carry out an analysis to determine the stress-strain behaviour at this critical location.

Determine the stress concentration factor

This is usually determined by finite methods. It is important to distinguish between the stress concentration factor (K _T ) and the fatigue strength reduction factor (K _F ). The fatigue strength reduction factor should be used in the fatigue analysis, although K _T may be used to calculate a conservative value of the fatigue crack initiation life.

Steps 3 and 4

Calculation of K _F

The effect of a notch on the fatigue strength of a component varies considerably with both notch geometry and material properties, and is usually less than that predicted by the use of K _T . Therefore, to model this effect the fatigue strength reduction factor must be calculated. This can be done, using Peterson's equation as described below (ref. ^[3] pp9-11). Firstly, the notch sensitivity, q, is defined as follows:

The notch sensitivity is dependent upon both the notch root radius, r, and a material parameter, a, such that:

This parameter appears to be a function of the tensile strength of the material and is calculated either from tabulated data or from empirical formulae. A number of values of a, for steels with various tensile strengths are given in Table 1.

Table 1 Values of the material parameter, a, for a number of steels with various tensile strengths (after ref ^[3] )

Equations [1] and [2] are combined to form the equation commonly known as Peterson's equation:

Step 5

Determine the monotonic and cyclic stress-strain properties

The monotonic properties required ( i.e. those obtained in an ordinary tensile test) are the Young's Modulus and the ultimate tensile strength of the material. The cyclic properties required are the cyclic strength coefficient (

) and the cyclic strain hardening exponent (

). A comparison of typical cyclic and monotonic properties for a medium strength steel is shown in Fig.3. The cyclic properties are determined by testing a smooth specimen under axial cyclic strain control. The cyclic stress-strain curve is then defined as the locus of tips of the stable hysteresis loops from several tests at different strain amplitudes, as shown in Fig.4. The equation defining the cyclic stress-strain curve is:

Given the above equation,

and

may be calculated from suitable test results.

Step 6

Determine material fatigue properties from smooth specimens

Another four empirically determined quantities are required to characterise the fatigue behaviour of a metal when using LSSA techniques. These are:

• The fatigue strength coefficient,

• The fatigue strength exponent, b

• The fatigue ductility coefficient,

• The fatigue ductility exponent, c

These properties can be obtained from the results of completely reversed strain-controlled tests, as described in Ref. ^[1] .

For each test, data for determining the stress and strain amplitude are taken from a hysteresis loop at approximately half of the fatigue life. Elastic strain is computed by extrapolating the elastic line of the hysteresis loop on the strain axis. Plastic strain is then computed by substracting the elastic strain from the total strain ( Fig.5).

When plotted on log/log axes, the elastic strain amplitude and plastic strain amplitude against reversals to failure form straight lines having the following equations:

Hence, by fitting straight lines to the data the required quantities can be obtained from [5] and [6] (Fig.6).

Step 7

Calculate stress and strain at notch root

To calculate the fatigue life of a real component we need some means of relating the local stress and strain to the remote ( i.e. nominal) stress and strain. The only method that can be realistically used for this requirement is known as 'Neuber's Rule'. This rule proposes separation of stress and strain by having two concentration factors instead of one. While one stress concentration factor is sufficient to control both stress and strain in the elastic region, this cannot be applied when strain is not proportional to stress. The separate concentration factors for stress and strain are defined as:

until yielding starts at the notch root, K σ = K ε (or K F ); after yielding, K σ decreases and K ε increases ( Fig.7). Neuber showed that the product (K σ.K ε) of the two concentration factors is constant, and it is this result which forms the basis of Neuber's Rule. From the definitions of K σ and K ε we have:

where

In the elastic region:

for both elastic and plastic local strains. Assuming that the nominal strain (remote from the notch) is elastic we can rewrite [11]:

Using [13] (Neuber's Rule) in conjunction with the appropriate cyclic stress-strain curve ([4]), a complete notch root strain history can be calculated from a given sequence of ΔS values. These are usually taken from a sequence of peaks and troughs representing service conditions. If two successive points are taken from such a sequence, a value of ΔS can be found, and thus for known values of K F and E, the right hand side of [13] evaluated. It is then necessary to find values of Δ σ and Δ ε which satisfy both [4] and [13], see Fig.8. These values must be calculated by an iterative method, as an analytical solution for the equations does not exist.

Figure 9 shows a typical stress-strain plot, as generated by applying a variable amplitude load to a notched specimen. This highlights two problems: firstly, how do we define the shape of a line joining two points on the plot ( e.g. 2-3, 3-4, etc); secondly, how do we account for the change in gradient which occurs when a stress-strain line rejoins a previous one?

The first problem is overcome by use of an assumption known as Massing's Hypothesis. With reference to Fig.9, this means that the curves forming the hysteresis loops ( e.g. 2-7, 10-11, 11-10, 7-2) have the same shape as the cyclic stress-strain curve for that material ([4]), but are twice its size. A section of the cyclic stress-strain curve is shown in Fig.9, between points 1 and 2.

The second problem means that as the loops are followed, a record must be kept of the changes from one part of the stress-strain curve to another when the loop closes. For this purpose a computational technique known as the 'Wetzel Availability Matrix' is commonly used. This is a technique which numerically integrates along a cyclic stress-strain curve to calculate fatigue damage. It requires that the history be divided into equal parts or 'bands,' which are then used to define the numerical value of the range and mean for each reversal. Each reversal is then defined in terms of line segments or elements, the lengths of which are equal to one band. Rules are then applied to the elements which ensure that the cyclic deformation response of the material is simulated.

Using these techniques a complete notch stress-strain history can be computed from a nominal stress-time history.

Step 8

Calculate fatigue damage

The equations which are commonly used to link life with strain separate the strain into elastic and plastic components. The unit of life considered in all local stress-strain damage calculations is one reversal, with what would normally be considered as one cycle consisting of two reversals.

Research during the 1950s resulted in the Manson-Coffin equation, which links the plastic strain range, Δ εp, to the life in cycles:

A similar equation is used when considering the elastic strain:

Morrow ^[4] proposed that the two components of strain be summed, resulting in the following equation:

Equation [16] provides a method for calculating the life to fatigue crack initiation from a given stress-strain history. However, it does not allow for any effect of mean stress on fatigue life.

Morrow suggested the following correction, which involves simply subtracting the mean stress σm, in [16]:

Another method for taking account of the mean stress in fatigue damage calculations is the Smith-Topper-Watson approach. This assumes that the maximum stress in a cycle, σ max , can be used as a measure of its position on the stress axis, and that there is a correlation between fatigue life and the product of the strain amplitude, ε a , and σ max resulting in the following equation:

Either [17] or [18] can be used to calculate the fatigue damage caused by any particular loading sequence; computational methods must be used to obtain a solution in a reasonable time.

The application of LSSA methods to welded joints

A number of research workers have attempted to apply these methods of fatigue analysis to welded joints, most of the work having been carried out at The University of Illinois. ^[5-7]

The fatigue life of any particular detail can be divided into two sections: crack initiation (which LSSA methods attempt to predict) and crack propagation. In some cases, particularly automobile and aircraft parts, the initiation life, that is the life required to produce a small crack of some nominal size, is regarded as the life to failure of the component. When this is not the case, fracture mechanics methods, as described for example in BS PD 6493, are used to calculate the crack propagation life. ^[5]

Estimation of K _F

As has been stated, a variety of data is required to perform the LSSA analysis. Perhaps the most critical piece of information is the fatigue strength reduction factor, K _F (see [1]-[3]).

Lawrence et al ^[5] introduced the concept of a critical value for the fatigue structure reduction factor. This is found by assuming a general form of an equation defining K _T :

where A, B and C are constants. Substituting this into Peterson's equation ([3]), differentiating K _F with respect to r, and setting that expression equal to zero leads to the result that K _F is a maximum when:

so that

The use of this concept when analysing welded joints is justified, in that it is virtually impossible to fix a value to the notch root radius in a location such as a weld toe.

Residual stress effects

The thermal strains which accompany welding induce residual stresses, which are usually assumed to be tensile and of yield point magnitude. These stresses influence the effective mean stress, an influence which may diminish with cycling.

From [17] and [18] it can be seen that a tensile mean stress can be expected to reduce fatigue life. In LSSA it is assumed that the mean stress reduces as fatigue continues; this is known as mean stress relaxation. Lawrence et al ^[6] propose use of the following equation (originally Jhansale and Topper):

where

If the stresses at the notch root can be considered to be essentially elastic, the damage per cycle is usually stated as:

allowing the life to initiation to be calculated from the following integral:

This equation allows the residual stresses due to welding to be taken into account. Lawrence et al [6] assume that the residual stress can take one of three values, + σ y , o or - σ y , which covers the most extreme possibilities, where σ y is the yield strength of the material in question.

Low cycle fatigue material data for weldments

There are few data available describing the cyclic properties of weldments and heat affected zones (HAZs). As most of the cyclic properties required in the LSSA are heavily dependent upon microstructure it is necessary to test specimens specific to the (predicted) fatigue crack initiation site being considered. [6] Some correlations between material hardness (as measured by BHN) and certain of the cyclic properties have been found [5,7] which provide a convenient means of determining these properties, the hardness being determined by microhardness measurements performed in the region of crack initiation.

Estimating the Initiated crack length

If the fatigue crack propagation life of the detail in question is to be calculated using a fracture mechanics approach, a good estimate of the initiated crack length (a I ) is required. A number of methods have been proposed, the most common being to assume that a I is 0.01in (approx. 0.25mm) in all cases. For cracks initiating from a weld defect such as a pore, slag inclusion or lack of penetration, a I is often taken as being equal to the relevant dimensions of the defect. Alternatively, some analytical methods for calculating a I have been proposed, with a I being dependent upon the weld geometry and size, and the ultimate tensile strength of the material.

Can LSSA methods be used in fatigue design?

With the increased availability of suites of computer programs for fatigue analysis of unwelded components, there is a temptation for the designer to attempt to apply these methods to welded joints. However, without the data necessary for such an analysis, such as precise descriptions of weld macro and micro-geometry and cyclic material properties at the point of crack initiation, the results are likely to be misleading. The designer is also faced with the problem of residual stresses, and in particular how the relaxation of these stresses should be treated in the analysis, if at all.

The effects on initiation life predictions of varying some of the local geometric and material parameters are considered below.

Effects of variable macro-geometry on local stress-strain analyses

For the majority of welded joints, it is impossible to make a precise prediction as to the macro-geometry of the finished weld. It is, of course, possible to set limits on this variation at the design stage, but one may well see considerable variation in the fatigue life predicted by local stress-strain methods for welds within these limits. For example, for a transverse butt weld (BS 5400 Class D or E), with weld angle varying from 20 to 60°, K Fcrit varies from 2.32 to 3.02. [5]

The following equation allows us to estimate 2N I (fatigue life to initiation) for long life fatigue (>10 [5] cycles), using local stress-strain parameters:

Using this equation, values of 2N _I have been calculated for various values of K _F , see Table 2 and Fig.10. The residual stress (σ_o ) is assumed to be of tensile yield magnitude. The values of

and b (fatigue strength coefficient and fatigue strength exponent) are for a weld HAZ in ASTM A36 grade structural steel.

The following (eq. [8] of Ref. ^[3] ) provides a method for calculating K _F :

where

t = plate thickness.

Table 2 Values of 2N I (fatigue life to Initiation) calculated using [25] for various values of K F , for the HAZ of a weld made In steel A36; with σ o = σ Y = 534 N/mm 2 , ΔS= 150 N/mm, σ f = 1192 N/mm 2 , b = -0.07

Table 3 Geometric coefficients for a full penetration double V butt weld ( Fig.11)

Table 4 K Fcrit for various weld toe angles in double V butt welds (t = 50mm)

Values of α for various joint types and geometries are given in Table 1 of Ref. [1] . Geometric coefficients for a full penetration double V butt weld (Fig.11) are given in Table 3.

From Fig.10 and 12 it can be seen that a considerable variation in K F and hence also in 2N I can be expected over relatively small ranges of weld toe angle, most noticeably for this particular geometry in the range 10° ≤ Ɵ ≤ 15° where the value of K F may vary from 1.1 to 2.5. The experimental data in Fig.13 ( Fig. 4.4 of Ref. [2] ) show less dependence of fatigue strength upon reinforcement angle than would be predicted using LSSA methods, however.

Effects of local variations in material properties

Consider the case of a fatigue. crack initiating at a weld toe. In such a position it is difficult to decide whether to use weld metal properties or HAZ properties in the fatigue calculations. The data in Table 5 show the variations in material properties for the various possible fatigue crack initiation sites shown in Fig.14.

Table 5 Cyclic and fatigue properties of base, weld, and heat affected materials for ASTM specification A514 welds using E110 filler (after ref. ^[5] )

Material	ASTM A514-BM	ASTM A514-HAZ	E110- WM(1P)	E110- WM(2P)
Cyclic yield strength, 0.2% offset,N/mm 2	604	938	649	603
Cyclic strain hardening exponent,	0.091	0.103	0.177	0.166
Cyclic strength coefficient, , N/mm 2	1090	1766	2022	1670
Fatigue strength coefficient, N/mm 2	1304	2001	1890	1408
Fatigue ductility coefficient,	0.975	0.783	0.848	0.595
Fatigue strength exponent, b	-0.079	-0.087	-0.115	-0.079
Fatigue ductility exponent, c	-0.699	-0.713	-0.734	-0.590
Transition fatigue life, 2N t , reversals	3461	1138	1536	6448

Using data from Table 5 and [25] values for the initiation life have been calculated for various nominal stress ranges ( Fig.15).

It can be seen that for a given weldment a wide range of fatigue crack initiation lives is predicted, depending on the material properties at the point of crack initiation.

This result is in conflict with a large body of data which shows that the fatigue life of welded joints is independent of material properties ( e.g. chapter 8 of Ref. ^[2] ). The fatigue strength at 2x10 ⁶ cycles of butt welds in a number of steels of different strengths is plotted in Fig.16; no correlation between fatigue life and material type can be seen.

Conclusions

A number of problems arise from the application of local stress-strain techniques to welded joints, mainly concerned with the variations of macro- and micro-geometry and of material parameters which are found in the region of a weldment. Use of these methods as a design tool must therefore be cast into some doubt, as the results are likely to be misleading.

It may be possible to apply these techniques to certain types of welded joint where the final conditions of the weldment are carefully controlled, but at present there is a lack of suitable data available to make such analyses.

Without such data, and at the present stage of development, it is felt that these methods cannot be justifiably applied to welded joints other than as a post-facto investigation.

References