MMLF calculates crack initiation life from data files collected by rosette strain gauges.

MMLF incorporates a Mròz-Garud cyclic plasticity model, together with a generalized form of the Wang-Brown multiaxial rainflow and damage accumulation procedures. The program also incorporates a variety of other critical plane damage models. The program requires measured strains from strain gauge rosettes as input, together with uniaxial cyclic material properties.

The software can handle general non-proportional biaxial loadings.

Uniaxial methods for life prediction using the local strain approach have been in use for some time, having their roots in the work of Basquin (Ref. 52), Manson (Ref. 53), and Coffin (Ref. 54)(Ref. 55), incorporating rainflow cycle counting and material memory and Miners rule (Ref. 57). Such methods are available in MSC Fatigue. Within the well known limitations of these methods they work quite well for a variety of components where the local loading in the critical area is uniaxial or near-uniaxial. This class of components includes many that are subject to complex multiaxial loading environments (Ref. 49)(Ref. 50). However, there are many other components where a combination of loads and geometric effects generates local loadings which are proportional or non-proportional multiaxial.

The life prediction process from measured strains can be divided into two steps. The first step is to determine the relationship between the measured strains and all the stress and strain components required for the damage calculation, through application of a cyclic plasticity model. The second step is to carry out cycle and damage accumulation. MMLF addresses problems where the strains can be measured with a rosette, i.e. biaxial loading on a free surface. For these problems the process can be summarized by the flow chart in Figure 6‑6:

The approach for calculations on the basis of elastic finite element analyses will be similar, except that the input strains will be elastic strain components from the surface of the structure, and a notch correction procedure will be required in addition to the cyclic plasticity model to estimate the elastic-plastic stresses and strains (Ref. 59).

The essential calculations made by MMLF are as follows:

These calculations are described in more detail in the following sections.

The stresses and strains required by the damage models can be calculated if the relation between the equivalent plastic strain increment and the equivalent stress increment is known during the application of a given load increment.

However, it is known that the current relation depends on the previous load path and therefore, the plasticity model must deal with loading path dependent material constitutive behavior.

Several models are available in the literature (Ref. 45), (Ref. 46), (Ref. 60), (Ref. 61) of which the model proposed by Mròz and recently modified by Garud are the most popular. Mròz has proposed that the uniaxial stress-strain material curve be represented by a set of plasticity surfaces in three dimensional stress space. In the case of a two dimensional stress state, the plasticity surfaces reduce to ellipses on the plane of principal stresses described by:

and illustrated in Figure 6‑7.

The load path dependent memory effects are modeled by prescribing a translation rule for the ellipses moving with respect to each other over distances given by the stress increments. It is also assumed that the ellipses move inside each other and they do not intersect. If the ellipses come in contact with one another they move together as a rigid body.

The translation rule proposed by Garud avoids the intersection of the ellipses that could occur in some cases in the original Mròz model. The Garud translation rule is illustrated in Figure 6‑8 and can be described by a model consisting, for simplicity, of only two plastic surfaces (ellipses).

In order to predict material response due to the stress increment dσ, the following steps are made:

The translation rule assures that the two ellipses are tangential with the common point B1B2 without intersecting each other. Two or more tangential ellipses translate as a rigid body and the largest moving ellipse (Figure 6‑8) indicates the proper constitutive relation (linear segment) to be used for a given stress increment. The principal of the method is described here for two surfaces. MMLF uses 40, with the corresponding linear sequences being fitted to the Ramberg-Osgood equation to give a more or less smooth curve. More details of this method can be found in (Ref. 47).

Wang and Brown (Ref. 62) proposed a multiaxial cycle counting method on the basis of strain hardening behavior under non-proportional variable amplitude loading. Relative stresses and strains were introduced so that a pair of turning points define the start and end points of a reversal, where the equivalent relative strain rises monotonically to a peak value. Since plastic deformation generates the driving force for small fatigue cracks, hysteresis hardening provides a physical parameter for cycle counting, analogous to rainflow counting in the uniaxial case.

Each reversal commences with elastic unloading, which is followed by reloading and plastic strain hardening up to the next turning point. The most significant turning point occurs at the highest value of equivalent strain. This is illustrated at time 0 in Figure 6‑9, which shows a repeating block of a combined tension/torsion non-proportional load history. The equivalent strain is defined as the von Mises strain.

The cycle counting method is illustrated by the following example. Starting from the most significant turning point, a graph is drawn for the loading block of relative equivalent strain, where relative strain represents the change of strain since time A. Figure 6‑10 shows the relative equivalent strain, with respect to times 0, 10 and 20 seconds. Using the relative strain, a reversal can be defined starting from 0, up to the maximum value 10 seconds. To obtain the second reversal the relative strain is re-plotted starting from the next turning point where unloading commences (at 10 seconds), and the portions of the strain hardening curve for the reversal are selected by a traditional rainflow procedure (Ref. 47). The region of unloading and reloading within that reversal is counted in the next step.

Using the next turning point, relative strain is replotted with respect to 20 seconds for the subsequent continuous fragment of strain history, yielding the third reversal. This procedure is repeated for each turning point in chronological order, until every fragment of strain history has been counted. The recursive nature of this process makes this module significantly slower than other analysis modules such as MCLF.

The counting method described above is independent of fatigue damage parameters, being based on hysteresis deformation behavior. Being unrelated to material properties, it can be integrated with any multiaxial fatigue damage model. If the counted reversals are non-proportional, a fatigue damage parameter that accounts for non-proportional straining effects is required. The path-independent damage parameter proposed by Wang and Brown (Ref. 63) has been shown to provide good correlation for several materials under proportional and non-proportional loading,

where γmax is the maximum shear strain amplitude on a critical plane (proportional or non-proportional), δεn is the normal strain excursion between the two turning points of the maximum shear strain (that is the range of normal strain experienced on the maximum shear plane over the interval from start to end of the reversal), and σn,mean is the mean stress normal to the maximum shear plane. This can be omitted if no mean stress correlation is required. If mean stress is included, this is equivalent to the Morrow method. The term S is a material constant determined from a multiaxial test (typically between 1 and 2 for Case A and around 0 for Case B) and v‘ is the effective Poisson’s ratio. The right hand side of the equation is the same as the uniaxial strain life equation, with a Morrow mean stress correction (Ref. 64). Mean stress is measured as the average of the maximum and minimum stress values over the reversal. The total damage induced by a loading history is calculated using Miner’s rule.

Figure 6‑11 is a plot of predicted life against experimental life for a variety of proportional and non-proportional tests on laboratory specimens, from (Ref. 48).

The other multiaxial damage parameters considered are the more conventional critical plane parameters. In these methods, stresses and strains are resolved onto a particular plane, inclined at an angle = 90 degrees (Case A) and/or = 45 degrees (Case B) to the free surface. Cycle counting (uniaxial) and damage parameter calculation is carried out on the critical plane and the damage accumulated. The orientation of the projection of the normal to the damage plane is increased by 10 degree increments from 0 to 170 degrees. The plane with the largest accumulated damage is said to be the critical plane. The models are:

MMLF may be used to predict the life of components and structures, to provide some help in design optimization, and to assist in the design of efficient tests through fatigue editing. The main drawback of the program is that it is necessary to place a strain gauge rosette on the critical location or crack initiation site in order to get a reliable prediction. This is due to the fact that there is no simple transfer function between the measured strains at a non-critical location and the strains at a nearby critical location when the loading is non-proportional.

Three applications of the multiaxial fatigue program will be described to illustrate the method.

This example is based on strain gauge measurements from an automotive wheel. The wheel was tested on a rig simulating the rotation of the wheel under load, and the strain gauge measurements were taken using a 45 degree rosette placed in the most critical location. This location was identified by previous stress analysis. A small section of the three strain gauge channels is illustrated in Figure 6‑12.

The three channels of strain are more or less sinusoidal with a phase difference between them, indicating that the stress state is multiaxial and non-proportional. The two cycles shown above are taken from a longer history. The stress-strain state variations can be better understood qualitatively by looking at the behavior of the principal stresses as illustrated in Figure 6‑13, where it can be seen that the two principal stresses are almost in phase, but with slightly different amplitudes and means. Notice that the absolute maximum principal stress (the principal stress with the largest magnitude) “flips” between the maximum and the minimum principals. Whenever it does this of course, the angle also flips through 90 degrees indicating that the state of stress has reached a condition where E.MAX and E.MIN are equal and opposite, i.e. pure shear. In fact the stress state behavior can be summed up by saying that when the stress is at a maximum or minimum, the stress state is almost equibiaxial, and in between times it passes through a condition of pure shear where the stresses are lower. Between the two limits the principal stress axis rotates through 90 degrees, so that there is a complete rotation (180 degrees) per cycle. These files were generated using SSA.

Another way of looking at this data is to plot the biaxiality ratio (the ratio of the smaller in magnitude to the larger principal stress) and the orientation against the largest principal stress. This is shown in Figure 6‑15 for 1 cycle.

The results of analysis of the full loading history are given below with the results of some other methods for comparison.

These results are all very similar, with the notable exception of the normal strain method which is relatively non-conservative.

It is normal to view the accumulated rainflow cycles in an analysis in the form of a 3D histogram plot. Normally the axes are range-mean or max-min or from-to. With the Wang-Brown method, each cycle is characterized by shear strain range, normal strain range, mean normal stress, and two angle and . The distribution of numbers of reversals and damage may reasonably be visualized in relation to any two of these parameters. Figure 6‑15 shows the distribution of reversals in relation to shear strain range and normal strain range.

Another potentially useful way of viewing the results is in the form of a polar plot of damage, as shown in Figure 6‑16. In this case there are only Case B damaging cycles, so there is only one curve on the plot. This plot was produced with the polar display program MPOD.

This example concerns a sign-off test which is carried out on a front suspension cross-member and steering rack assembly. The steering rack is mounted to the cross-member by means of two small brackets, which are fatigue-sensitive areas. The assembly is tested by fixing the cross-member and applying a constant amplitude, unidirectional load to the steering rack. The resulting stress state variations are illustrated in Figure 6‑17.

These plots show that the single-axis loads applied generate locally a near-uniaxial, near proportional stress state. The results of analysis with the new program, assuming 99.9% certainty of survival (based on the calculated standard errors on the material parameters) are as given in below.

This example illustrates how the program can be used to reduce testing times through multiaxial, multichannel fatigue editing of simulation test rig drive files. The component in question is a rear axle from an off-road vehicle. The axle is instrumented with three strain gauge rosettes close to critical locations, and analysis of the measurements from all three strain gauge rosettes show that the loading is non-proportional. It is desired to reduce the testing time by editing the rig drive signals while retaining all the sections of the original sections that cause significant damage. This was carried out using a simple editing procedure.

One of the outputs of the life prediction program is a time-correlated damage file, essentially a time history of damage. This file is a record of the damage accumulated, with damage for each reversal being distributed between the opening and closing point of each reversal. The signal is divided up into a number of equally spaced windows, and the least damaging are discarded until a certain percentage of damage is retained. In this example, 95% was retained. The time slices to be retained are compared across the three channels with a logical OR operation and all the resulting windows are retained. The resulting edit vector can be used to edit the rig drive signals, inserting suitable joining functions for continuity. It is also important to edit the strain signals and recalculate the life in order to check the likely effect of editing on the fatigue damage. This is important because of the possibility of reversals starting in one window and ending in another, and also because in multiaxial loading the loading path is important as well as the peaks. In some cases, editing the signals may actually cause a slight increase in fatigue damage. This is why methods such as described in (Ref. 68) are not really suitable for non-proportional loadings because they do not retain the important sections of the loading path. The results of the analysis described are summarized below and the original and edited strain signals for Gauge 1 are illustrated in Figure 6‑18.

The advantages of this method are that it retains both the most important sections of loading path, and the essential frequency content of the loading, essential for components with dynamic behavior.

The MMLF module can be run in one of the following 3 modes:

Once running in interactive mode the MMLF module will display the following screen.

The user must enter the name of an Input Job File, the extension .fjb will be appended automatically. If the name of an existing Job File is entered then MMLF will go to the postprocessing menu screen. If a new job is specified then its parameters have to be entered in a series of windows. If no name is entered then the results of processing may NOT be saved.

If a new job is specified, its name is specified in this window and its parameters in subsequent windows.

The flow chart below illustrates the major routes through MMLF.

The above flowchart is mirrored by the options that are on the Post-processing menu illustrated in Figure 6‑31. If a new job is being defined, then the following screen will appear.

The Service Loading fields are as follows:

The fields are as follows:

The damage accumulation methods are:

Regarding Miner’s constant, operation at a strain amplitude will result in failure in, say, N1 cycles. Operation at the same strain amplitude for a number of cycles less than N1, Nj say, will result in a smaller fraction of damage, Dj, which is often referred to as a partial damage. Operation over a spectrum of different strain levels, results in a partial damage contribution Di from each strain cycle,εi. Failure is then predicted when the sum of these partial damage fractions reaches unity so that:

D1 + D2 +...... + Di-1 + Di ≥ 1

The Palmgren-Miner rule asserts that the partial damage at any strain amplitude , or STW level, is linearly proportional to the ratio of number of cycles of operation, ni, to the total number of cycles that would produce failure at that strain level, Ni i.e.:

Di = ni / Ni

Failure is predicted then if:

or

The above equations represent a statement of the linear damage rules used by all the MSC Fatigue analyzers. Experience shows that linear damage summation is somewhat of an over-simplification of reality. The most important shortcoming is that no account is made of the sequence in which strain levels are experienced, and damage is assumed to accumulate at the same rate for a given strain level regardless of prehistory. In particular, it appears that large strain amplitudes which precede smaller ones, cause the smaller cycles to become more damaging than expected and visa versa. The net result is that in the first instance, the Miners' sum (total of (n/N)) is measured to be less than 1 and in the latter case, it becomes greater than 1. Since most service environments involve quasi-random loading sequences, the use of the Palmgren-Miner linear damage rule summing to a constant of 1 is mostly satisfactory. However, provisions have been made within MMLF to set the Miners' sum to any desired value.

The multiaxial local stress-strain fatigue analyzer requires a characterization of the material from which the component or part is manufactured. This characterization is accomplished through a set of parameters generated by strain-controlled testing smooth specimens of the material in question.

This screen allows selection of material data-sets to be made. It is possible to either load the desired parameters from the PFMAT data base, to load them from a user custom database (any name allowed) or to enter them directly from the keyboard, or to generate them from empirical rules based on tensile properties.

Regarding surface finish and treatment: the effect of surface treatment depends on the surface finish. The increase in endurance limit with stress due to the surface treatment is given below.

Whatever correction was made by the surface finish, then applying a surface treatment will have a subsequent effect taken from the table above. For example if machining reduces the endurance limit by 50%, and it is wished to recover the loss, then from the table it can be seen that cold rolling will increase the limit by +70%.

The output files generated by any of the combination of settings to which MMLF can be set are shown below. Note that the files generated in any given run of MMLF depend upon the parameters specified in the job file.

MMLF can generate a number of different output files depending on the analysis path chosen. If a multiple analysis has been specified in any of the input screens, then a single file, with the results name provided with any one of the following file extensions, depending on the multiple parameter, will be created.

All these files have the standard X-Y paired data format and can be displayed graphically within MMLF itself or by the X-Y parameter display module, MTPD. No other output files will be created.

If, on the other hand, a single shot analysis has been carried out then MMLF can optionally create a group of files which contain different information, have the same results name and the following file extensions.

In addition, numerical results are written to the extra details area of the damage matrix file, .dhh extension. The following table details the keywords and values stored.

The fatigue life represents the number of repeats of the history to failure. In this context “failure” is taken to be that defined to be the initiation of an 'engineering crack', typically 2mm. The number of cycles or reversals used corresponds to the number of cycles or reversals in the strain record.

If a multiple analysis procedure has been specified then, on completion, MMLF will display a table of results on the screen and make a similar entry to the notebook and generate an X-Y file of results. In the case of a certainty of survival analysis, entering ALL in response to the question, and introducing scatter (e.g. SD=0.2 in the materials data edit screen), will cause the following table to be displayed:

An example of an X-Y plot is shown below.

The general module structure flowchart in Figure 6‑20 at the start of the Module Operation section illustrates how MMLF will always lead the user to the Postprocessing Menu.

The Postprocessing menu looks like this:

The menu shown above can be mapped directly to the flow chart in Figure 6‑20.

The Postprocessing options screen will be reached when:

The Postprocessing menu allows a job to have one or more of its parameters changed, and to have existing results and parameters changed and/or viewed.

The options on the menu that allow current inputs to be altered are:

The other options may or may not affect the results of the calculation, but are generally of the nature of utilities rather than calculation parameters. They are as follows:

This displays the results of the most recently calculated life estimation. Exactly what is available for display depends upon the calculation; if an X-Y file was not created then it cannot be displayed, a multiple calculation will not produce any histogram files, but will produce an X-Y plot, etc.

Essentially this option plots the 7 non-zero stress and strain components. These are:

.sgx - σx(t), .sgy - σy(t), .txy - σxy(t), .epx - εx(t), .epy - εx(t), .epz - εz(t), .gxy - γxy(t)

The full range of possibilities is explained below:

The Job utilities menu has three options:

The personal preferences menu allows some aspects of MMLF's operation to be customized. These are:

Allows a value to be entered for the life of the back calculation.

In order to see the results of any change to the parameters in a job file, this option must be selected. Until this option is selected any changes will be ignored, and the job and results files will not be updated.

Exits from MMLF.

MMLF writes or reads the following entries to or from the local user environment.

It is recommended that, by default, the /OV=Y keyword be included in every batch command line, since if it is omitted and an output file with the specified name already exists, batch operation will cease. To finish an MMLF batch line use /OPT=X to exit from the postprocessing screen.

When batch processing with a series of different inputs, it is necessary to use a new batch line and option definition for each new input. The new line must specify the option from the postprocessing menu into which the new input will go.

A list of MMLF’s batch keywords: