S-N Curve

The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure. Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).

Figure 1: S-N data from testing

When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude Sa or range SR versus cycles to failure N, the relationship between S and N can be described by straight line segments. Normally, a one or two segment idealization is used.

Figure 2: One segment S-N curves in log-log scale

for segment 1

(1)

Where, S is the nominal stress range, Nf  are the fatigue cycles to failure, b1 is the first fatigue strength exponent, and S1 is the fatigue strength coefficient.

The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as high-cycle fatigue.

S-N curve data is provided for a given material on a MATFAT Bulk Data entry. It is referenced through a Material ID (MID) which is shared by a structural material definition.

Cycle Counting

Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence. One way to understand “cycle counting” is as a changing stress-strain versus time signal. Cycle counting will count the number of stress-strain hysteresis loops and keep track of their range/mean or maximum/minimum values.

Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points (S1, S2, and S3) will define two consecutive ranges as S1 = |S1 - S2| and S2 = |S2 - S3|. A cycle from S1 to S2 is only extracted if S1 S2. Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.

Figure 3: Determine cycles using rainflow cycle counting method

Parameters affecting rainflow cycle counting may be defined on a FATPARM Bulk Data entry. The appropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

Equivalent Nominal Stress

Since S-N theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the S-N curve.

Various stress combination types are available with the default being “Absolute maximum principle stress”. “Absolute maximum principle stress” is recommended for brittle materials, while “Signed von Mises stress” is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value.

Parameters affecting stress combination may be defined on a FATPARM Bulk Data entry. The appropriate FATPARM Bulk Data Entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

Damage Model (Mean Stress Influence)

Generally S-N curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully reversed and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to take into account the effect of non-zero mean stresses.

The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:

Gerber:

(2)

Goodman:

(3)

Where,

is the Mean stress given by

is the Stress Range given by

Se is the stress range for fully reversed loading that is equivalent to the load case with a stress range Sr and a mean stress

Su is the ultimate strength

The Gerber method treats positive and negative mean stress correction in the same way that mean stress always accelerates fatigue failure, while the Goodman method ignores the negative means stress. Both methods give conservative result for compressive means stress. The Goodman method is recommended for brittle material while the Gerber method is recommended for ductile material. For the Goodman method, if the tensile means stress is greater than UTS, the damage will be greater than 1.0. For the Gerber method, if the mean stress is greater than UTS, the damage will be greater than 1.0, with either tensile or compressive.

A Haigh diagram characterizes different combinations of stress amplitude and mean stress for a given number of cycles to failure.

Figure 4: Haigh diagram and mean stress correction methods

Parameters affecting mean stress influence may be defined on a FATPARM Bulk Data entry. The appropriate FATPARM bulk data entry may be referenced from a fatigue subcase definition through the FATPARM Subcase Information entry.

Damage Accumulation Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:

(4)

Where, Nif is the materials fatigue life (number of cycles to failure) from its S-N curve at a combination of stress amplitude and means stress level i , ni is the number of stress cycles at load level i , and Di is the cumulative damage under ni load cycle.

The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage, due to cyclic fatigue loading. However, it has been proved to work well for many applications.

Strain-life analysis is based on the fact that many critical locations such as notch roots have stress concentration, which will have obvious plastic deformation during the cyclic loading before fatigue failure. Thus, the elastic-plastic strain results are essential for performing strain-life analysis.

Neuber Correction

Neuber correction is the most popular practice to correct elastic analysis results into elastic-plastic results.

In order to derive the local stress from the nominal stress that is easier to obtain, the concentration factors are introduced such as the local stress concentration factor , and the local strain concentration factor .

(5)

(6)

Where, is the local stress, is the local strain, S is the nominal stress, and e is the nominal strain. If nominal stress and local stress are both elastic, the local stress concentration factor is equal to the local strain concentration factor. However, if the plastic strain is present, the relationship between and no long holds. Thereafter, focusing on this situation, Neuber introduced a theoretically elastic stress concentration factor Kt defined as:

(7)

Substitute Eq.5 and Eq.6 into Eq.7, the theoretical stress concentration factor Kt can be rewritten as:

(8)

Through linear static FEA, the local stress instead of nominal stress is provided, which implies the effect of the geometry in Eq.8 is removed, thus you can set Kt as 1 and rewrite Eq.8 as:

(9)

Where, e ,  is locally elastic stress and locally elastic strain obtained from elastic analysis, , the stress and strain at the presence of plastic strain. Both and can be calculated from Eq.9 together with the equations for the cyclic stress-strain curve and hysteresis loop.

Monotonic Stress-Strain Behavior

Relative to the current configuration, the true stress and strain relationship can be defined as:

(10)

(11)

Where, A is the current cross-section area, l  is the current objects length, l0 is the initial objects length, and and are the true stress and strain, respectively, Figure 5 shows the monotonic stress-strain curve in true stress-strain space. In the whole process, the stress continues increasing to a large value until the object fails at C.

Figure 5: Monotonic stress-strain curve

The curve in Figure 5 is comprised of two typical segments, namely the elastic segment OA and plastic segment AC. The segment OA keeps the linear relationship between stress and elastic strain following Hooke Law:

(12)

Where, E is elastic modulus and is elastic strain. The formula can also be rewritten as:

(13)

by expressing elastic strain in terms of stress. For most of materials, the relationship between the plastic strain and the stress can be represented by a simple power law of the form:

(14)

Where, is plastic strain, K is strength coefficient, and n is work hardening coefficient. Similarly, the plastic strain can be expressed in terms of stress as:

(15)

The total strain induced by loading the object up to point B or D is the sum of plastic strain and elastic strain:

(16)

Cyclic Stress-Strain Curve

Material exhibits different behavior under cyclic load compared with that of monotonic load. Generally, there are four kinds of response.

Which response will occur depends on its nature and initial condition of heat treatment. Figure 6 illustrates the effect of cyclic hardening and cyclic softening where the first two hysteresis loops of two different materials are plotted. In both cases, the strain is constrained to change in fixed range, while the stress is allowed to change arbitrarily. If the stress range increases relative to the former cycle under fixed strain range, as shown in the upper portion of Figure 6, it is called cyclic hardening; otherwise, it is called cyclic softening, as shown in the lower portion of Figure 6. Cyclic response of material can also be described by specifying the stress range and leaving strain unconstrained. If the strain range increases relative to the former cycle under fixed stress range, it is called cyclic softening; otherwise, it is called cyclic hardening. In fact, the cyclic behavior of material will reach a steady state after a short time which generally occupies less than 10 percent of the material total life. Through specifying different strain ranges, a series of hysteresis loops at steady state can be obtained. By placing these hysteresis loops in one coordinate system, as shown in Figure 7, the line connecting all the vertices of these hysteresis loops determine cyclic stress-strain curve which can be expressed in the similar form with monotonic stress-strain curve as:

Figure 6: Material cyclic response (a) Cyclic hardening; (b) Cyclic softening

Figure 7: Definition of stable stress-strain curve

(17)

Where, K' is the cyclic strength coefficient and n' is the strain cyclic hardening exponent.

Hysteresis Loop Shape

Bauschinger observed that after the initial load had caused plastic strain, load reversal caused materials to exhibit anisotropic behavior. Based on experiment evidence, Massing put forward the hypothesis that a stress-strain hysteresis loop is geometrically similar to the cyclic stress strain curve, but with twice the magnitude. This implies that when the quantity is two times of , the stress-strain cycle will lie on the hysteresis loop. This can be expressed with formulas:

(18)

(19)

Expressing in terms of , in terms of , and substituting it into Eq. 17, the hysteresis loop formula can be calculated as:

(20)

Almost a century ago, Basquin observed the linear relationship between stress and fatigue life in log scale when the stress is limited. He put forward the following fatigue formula controlled by stress:

(21)

Where, a is the stress amplitude, is the fatigue strength coefficient, and b is the fatigue strength exponent. Later in the 1950s, Coffin and Manson independently proposed that plastic strain may also be related with fatigue life by a simple power law:

(22)

Where, is the plastic strain amplitude, is the fatigue ductility coefficient, and c is the fatigue ductility exponent. Morrow combined the work of Basquin, Coffin and Manson to consider both elastic strain and plastic strain contribution to the fatigue life. He found out that the total strain has more direct correlation with fatigue life. By applying Hooke Law, Basquin rule can be rewritten as:

(23)

Where, is elastic strain amplitude. Total strain amplitude, which is the sum of the elastic strain and plastic stain, therefore, can be described by applying Basquin formula and Coffin-Manson formula:

(24)

Where, is the total strain amplitude, the other variable is the same with above.

Figure 8: Strain-life curve in log scale

Damage Models (Mean Stress Influence)

The fatigue experiments carried out in the laboratory are always fully reversed, whereas in practice, the mean stress is inevitable, thus the fatigue law established by the fully reversed experiments must be corrected before applied to engineering problems.

Morrow

Morrow is the first to consider the effect of mean stress through introducing the mean stress 0 in fatigue strength coefficient by:

(25)

Thus, the entire fatigue life formula becomes:

(26)

Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain.

Smith, Watson and Topper

Smith, Watson and Topper proposed a different method to account for the effect of mean stress by considering the maximum stress during one cycle (for convenience, this method is called SWT in the following). In this case, the damage parameter is modified as the product of the maximum stress and strain amplitude in one cycle.

                 (27)

The SWT method will predict that no damage will occur when the maximum stress is zero or negative, which is not consistent with the reality.

When comparing the two methods, the SWT method predicted conservative life for loads predominantly tensile, whereas, the Morrow approach provides more realistic results when the load is predominantly compressive.

Damage Accumulation Model

In the E-N approach, use the same damage accumulation model as the S-N approach, which is Palmgren-Miner's linear damage summation rule.

Experiments show that cracks nucleate and grow on specific planes known as critical planes. The Critical Plane Approach captures the physical nature of damage in its damage assessment process. It deals with stresses and strains on the critical planes. Depending on the material and stress states, the critical planes can be either maximum shear planes or maximum tensile stress planes. Therefore, in order to assess damage from multiaxial loads, two separate damage models are required. One is for crack growth due to shear, and the other is for crack growth due to tension.

Figure 15: Cracks driven by shear and tensile stress

Figure 16: Cyclic Torsion, Shear Strain, Tensile Strain, Shear and Tensile Damage.

Any damage model can be used in the critical plane approach. The damage models require a search for the most damaging plane. There are two possible damaging (or failure) modes. One on planes that are perpendicular to the free surface which is tensile crack growth. The angle θ is the angle that a crack is observed on the surface relative to the σx direction. The second failure system occurs on planes oriented 45 degrees to the surface, which is shear crack growth. Both in-plane and out-of-plane shear stresses are considered on this plane. θ can take on any value on the surface. The shear stress A is an in-plane shear stress and causes microcracks to grow along the surface. The maximum out-of-plane shear, B occurs on a plane that is oriented at 45 degrees from the free surface and causes microcracks to grow into the surface.

Figure 17: In–plane shear stress and normal stress at 90 degree plane; in-plane and out-of-plane shear stress at a 45 degree plane.

OptiStruct searches for the most damaging plane by 10 degrees of θ. On each plane, OptiStruct assesses damage using tensile crack damage model and shear crack damage model. At the end of a search, OptiStruct reports damage at the most damaging plane which is a critical plane.

The Stress-Life Approach for the Multiaxial Fatigue Analysis is similar to Uniaxial Fatigue Analysis. See the S-N Curve and Cycle Counting sections of Uniaxial Fatigue Analysis for introductory information for Stress-Life approach in Multiaxial Fatigue Analysis.

Damage models

Depending on the material, stress state, environment, and strain amplitude, fatigue life will usually be dominated either by microcrack growth along shear planes or along tensile planes. Critical plane models incorporate the dominant parameters governing either type of crack growth. Due to the different possible failure modes, shear or tensile dominant, no single damage model should be expected to correlate test data for all materials in all life regimes. There is no consensus yet as to the best damage model to use for multiaxial fatigue life estimates. Multiple models are used in OptiStruct multiaxial fatigue analysis. For stress-based damage model, Goodman model is used for tensile crack. Findley model is used for shear crack. You can define damage models after the MDMGMDL field on the FATPARM Bulk Data Entry. If multiple damage models are defined, OptiStruct selects the worst damage model from all the available damages.

Goodman model

The Goodman model in uniaxial fatigue analysis is used at critical plane to assess damage caused by tensile crack growth.

Findley model

The Findley criterion is often applied for finite long-life fatigue.

     (28)

Where, is computed from the torsional fatigue strength coefficient, , using:

           (29)

The correction factor typically has a value of about 1.04. Note that has to be defined based on amplitude. If is not defined by you, OptiStruct automatically calculates it using the following equation.

The constant k is determined experimentally by performing fatigue tests involving two or more stress states. For ductile materials, k typically varies between 0.2 and 0.3. Its default value in OptiStruct is 0.3.

If CHK is set to YES in FATPARM for the proportional loadcase, the above equation is calculated on maximum shear stress plane. For more details, see Proportional Biaxial Load.

Figure 18: Analysis flow for stress based multiaxial fatigue analysis

If the applied load is non-proportional multiaxial cyclic, OptiStruct runs the Jiang-Sehitoglu plasticity model to calculate the total strain and elasto-plastic stress. The Jiang-Sehitoglu model is one of the more successful incremental multiaxial cyclic plasticity models (Jiang and Sehitoglu "Modeling of Cyclic Ratcheting Plasticity, Part I: Development of Constitutive Equations," Journal of Applied Mechanics, Vol. 63, 1996, 720-725). In OptiStruct, isotropic hardening part is removed from Jiang-Sehitoglu's original model, which is best described in deviatoric stress space.

    (31)

Yield function

The Mises yield function, F, is expressed as:

   (32)

The notation is used for the deviatoric stress tensor, is the backstress tensor and k is the yield stress is simple shear.

Flow rule

The normality rule is given by:

   (33)

Where, is the exterior unit normal to the yield surface at the loading point, as defined below:

      (34)

Hardening rule

The total backstress is divided into M parts.

                   (35)

The evolution of the backstress (translation of the yield surface) for each of the parts is given by:

   (36)

Where, dp is the equivalent plastic strain increment and ci, ri, and Xi are three sets of non-negative and single valued scalar functions:

Material memory is contained in the α terms. The plastic modulus function, H, is derived from the consistency condition which requires that the stress state be on the yield surface when plastic deformation is occurring.

    (40)

There are four material parameters in this model, c, r, X, and k. All of these constants are computed from the cyclic stress strain curve of the material (Eq. 17). A simple power function is fit to this curve to obtain three material properties; cyclic strength coefficient, K', cyclic strain hardening exponent, n', and elastic modulus, E.

The shear yield strength, k, is obtained by setting the plastic strain to 0.0002 (0.02%) and dividing by . Both c and r are obtained by selecting a series of stress strain pairs along the material cyclic stress strain curve and describe the shape of the curve. Ratcheting rate is controlled by X which is set at a fixed value of 5. The number of components (M) is set to 10.

Non-proportional Hardening

Non-proportional hardening is the term used to describe loading paths where the principal strain axes rotate during cyclic loading. The simplest example is a bar subjected to alternating cycles of tension and torsion loading. Between the tension and torsion cycles the principal axis rotates 45 degrees. Out-of-phase loading is a special case of non-proportional loading and is used to denote cyclic loading histories with sinusoidal or triangular waveforms and a phase difference between the loads. Materials show additional cyclic hardening during this type of loading that is not found in uniaxial or any proportional loading path.

The 90 degrees out-of-phase loading path has been found to produce the largest degree of non-proportional hardening. The magnitude of the additional hardening observed for this loading path as compared to that observed in uniaxial or proportional loading is highly dependent on the microstructure and the ease with which slip systems develop in a material. A non-proportional hardening coefficient, a, can be introduced which is defined as the ratio of equivalent stress under 90 degrees out-of-phase loading/equivalent stress under proportional loading at high plastic strains in the flat portion of the stress-strain curve. This term reflects the maximum degree of additional hardening that might occur for a given material. OptiStruct measures the non-proportional hardening coefficient, a, at 0.003 strain amplitude using:

            (41)

Where,

You will need to input the coefkp90 and coefnp90. The default value of coefkp90 is 1.2. Default value of coefnp90 is 1.0.

The degree of non-proportionality, C, is a fourth ranked tensor which is a function of the plastic strain and the direction of the plastic strain increment.

   (42)

The amount of non-proportional hardening is computed from Tanaka's parameter. (Tanaka, E., "A Non-proportionality Parameter and a Cyclic Viscoplastic Constitutive Model Taking into Account Amplitude Dependencies and Memory Effects of Isotropic Hardening," European Journal of Mechanics, A/Solids, Vol. 13, 1994, 155-173).

        (43)

For any proportional loading A=0 and for 90 degrees out-of-phase loading A = 1/sqrt(2). The material constants, ri and the yield stress k are related to the non-proportional hardening parameter as follows:

The rate of hardening is controlled by the constant b. Since you are interested in the stable solution for fatigue calculations, b = 5 is selected for numerical stability. These equations are now sufficient to solve for the stresses and strains under any arbitrary loading path. The solution now proceeds by incrementally solving these equations, which can be solved in either stress, or strain control. The initial conditions are that the stress and backstress start at. All material constants start at their initial values as well and are updated after each loading increment.

Notch Correction

In uniaxial loading approximate solutions such as Neuber's rule are used to compute the stresses and strains during plastic deformation. Unfortunately, Neuber's rule cannot be directly extended to multiaxial loading because there are six unknowns and only five equations. To overcome this problem Koettgen (Koettgen V.B., Barkey M.E., and Socie, D.F. "Pseudo Stress and Pseudo Strain Based Approaches to Multiaxial Notch Analysis" Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, No. 9, 1995, 981-1006) proposed a structural yield surface to obtain local elastic-plastic stresses and strains. Having defined a "yield surface," standard cyclic plasticity methods can be used to solve the unknown stresses and strains at the notch. The material memory effects are built directly into standard cyclic plasticity calculations. The method is based on the same concept at the analytical or experimental nominal stress - notch strain curves used in uniaxial fatigue analysis such as the one shown below.

Cyclically, nominal stress - notch root strain response has all the features associated with stress-strain response such as hysteresis, memory, cyclic hardening and softening. The concept of pseudo stress, , or strain, , is defined as theoretical elastic stress or strain computed using elastic assumptions. Neuber's rule may be written in terms of pseudo stress as:

                (45)

These concepts can be generalized to three dimensional stress and strain states with a structural yield surface at a notch. Having defined a structural yield surface, standard cyclic plasticity methods can be used to solve the unknown stresses and strains at the notch. For multiaxial loading, the total stress or strain is obtained by superposition of the individual load cases. A relationship between pseudo stress and notch strain is described with a power function that has the same form as a stress-strain curve.

           (46)

Where, the constants K* and n* represent the behavior of the structure rather than the material. Uniaxial Neuber's rule is employed to establish the constants E*, K* and n*.

For plane stress on the surface, , and are all zero and this yield function can be written as:

       (47)

This equation defines a structural yield function, Fo, which has the same form as the yield functions used in the plasticity models for stress-strain calculations, i.e. a Mises yield function in terms of pseudo stress.

       (48)

The process to calculate stress from pseudo stress is summarized as follows:

Proportional Biaxial Load

If applied load is proportional biaxial load, you do not have to calculate incremental plastic stress and strain using Jiang-Sehitoglu plasticity model. Because direction of principal stresses do not change, you know where the maximum shear stress and the maximum tensile stress are developed and what their values are with Mohr's circle. What you need to do is convert pseudo stress range to stress range using notch correction method such as Neuber's rule or Glinka's rule. OptiStruct applies Hoffmann-Seeger method for this situation.

If only one static subcase is referenced in FATEVNT, the cyclic load is always proportional biaxial load. In this case, if CHK is set to YES in FATPARM, OptiStruct runs Hoffmann-Seeger method to calculate stress and strain from pseudo stress. If multiple static subcases are referenced in FATEVNT, currently OptiStruct runs Jiang-Sehitoglu plasticity model.

The assumptions within the Hoffmann-Seeger method are:

The steps below outline how OptiStruct calculates stresses and strains based on nominal stress using Hoffmann-Seeger method.

    (49)

2.	The equivalent strain and stress and is obtained by solving the following equations:

                   (50)

                                (51)

Plastic portion of the equivalent strain:

                                   (52)

                             (53)

       (54)

The same procedure can be applied to calculate the amplitude of each quantity assuming Massing assumption is valid for equivalent stress and strain.

Where, and are obtained by:

                                   (55)

     (56)

Range of each quantity can be obtained by doubled amplitude.

Damage models

Depending on the material, stress state, environment, and strain amplitude, fatigue life will usually be dominated either by microcrack growth along shear planes or along tensile planes. Critical plane models incorporate the dominant parameters governing either type of crack growth. Due to the different possible failure modes, shear or tensile dominant, no single damage model should be expected to correlate test data for all materials in all life regimes. There is no consensus yet as to the best damage model to use for multiaxial fatigue life estimates. Multiple models are used in OptiStruct multiaxial fatigue analysis. For strain-based damage model, one model for tensile crack growth, Smith-Watson-Topper is used and two models for shear crack growth, Fatemi-Socie model and Brown-Miller model are used. The user can define damage models after the MDMGMDL field on the FATPARM Bulk Data Entry. If multiple damage models are defined, OptiStruct selects the worst damage model from all the available damages.

Smith-Watson-Topper model

This model is for tensile crack growth. In very high strength steels or 304 stainless steel under some local histories, cracks nucleate in shear, but fatigue life is controlled by crack growth on planes perpendicular to the maximum principal stress and strain. Both strain range and maximum stress determine the amount of fatigue damage.

Figure 19: SWT damage model

This model, commonly referred to as the SWT parameter, was originally developed and continues to be used as a correction for mean stresses in uniaxial loading situations. The SWT parameter is used in the analysis of both proportionally and non-proportionally loaded components for materials that fail primarily due to tensile cracking. The SWT parameter for multiaxial loading is based on the principal strain range, Δε1 and maximum stress on the principal strain range plane, .

   (57)

The stress term in this model makes it suitable for describing mean stresses during multiaxial loading and non-proportional hardening effects.

Fatemi-Socie model

This model is for shear crack growth. During shear loading, the irregularly shaped crack surface results in frictional forces that will reduce crack tip stresses, thus hindering crack growth and increasing the fatigue life. Tensile stresses and strains will separate the crack surfaces and reduce frictional forces. Fractographic evidence for this behavior has been obtained. Fractographs from objects that have failed by pure torsion show extensive rubbing and are relatively featureless in contrast to tension test fractographs where individual slip bands are observed on the fracture surface.

Figure 20: Fatemi-Socie model

To demonstrate the effect of maximum stress, tests with the six tension-torsion loading histories were conducted, that were designed to have the same maximum shear strain amplitudes. The cyclic normal strain is also constant for the six loading histories. The experiments resulted in nearly the same maximum shear strain amplitudes, equivalent stress and strain amplitudes and plastic work. The major difference between the loading histories is the normal stress across the plane of maximum shear strain.

The loading history and normal stress are shown in the figure at the top of each crack growth curve. Higher maximum stresses lead to faster growth rates and lower fatigue lives. The maximum stress has a lesser influence on the initiation of a crack, if crack initiation is defined on the order of 10 mm, which is the size of the smaller grains in this material.

These observations lead to the following damage model that can be interpreted as the cyclic shear strain modified by the normal stress to include the crack closure effects.

     (58)

The sensitivity of a material to normal stress is reflected in the value k/σy. Where, σy is stress where a significant total strain of 0.002 is used in OptiStruct. If test data from multiple stress states is not available, k = 0.3. This model not only explains the difference between tension and torsion loading, but also can be used to describe mean stress and non-proportional hardening effects. Critical plane models that include only strain terms cannot reflect the effect of mean stress or strain path dependent on hardening.

If CHK is set to YES in FATPARM for proportional loadcase, (Eq. 58) is calculated on the maximum shear strain plane. For more details, see Proportional Biaxial Load. If or are not available, OptiStruct calculates them using the following relationship. The transition fatigue life, 2Nt, is selected because the elastic and plastic strains contribute equally to the fatigue damage. It can be obtained from the uniaxial fatigue constants.

         (59)

The Fatemi-Socie model can be employed to determine the shear strain constants.

      (60)

First, note that the exponents should be the same for shear and tension.

                 (61)

Shear modulus is directly computed from the tensile modulus.

      (62)

Yield strength can be estimated from the uniaxial cyclic stress strain curve.

           (63)

Normal stresses and strains are computed from the transition fatigue life and uniaxial properties.

    (64)

Substituting the appropriate the value of elastic and plastic Poisson’s ratio gives:

Separating the elastic and plastic parts of the total strain results in these expressions for the shear strain life constants:

Brown-Miller model

This model is for shear crack growth. Brown and Miller conducted combined tension and torsion tests with a constant shear strain range. The normal strain range on the plane of maximum shear strain will change with the ratio of applied tension and torsion strains. Based on the data shown below for a constant shear strain amplitude, Brown and Miller concluded that two strain parameters are needed to describe the fatigue process because the combined action of shear and normal strain reduces fatigue life.

Figure 21: Fatigue Life vs Normal Strain Amplitude

Influence of normal strain amplitude

Analogous to the shear and normal stress proposed by Findley for high cycle fatigue, they proposed that both the cyclic shear and normal strain on the plane of maximum shear must be considered. Cyclic shear strains will help to nucleate cracks and the normal strain will assist in their growth. They proposed a simple formulation of the theory:

       (67)

where, is the equivalent shear strain range and S is a material dependent parameter that represents the influence of the normal strain on material microcrack growth and is determined by correlating axial and torsion data. is the maximum shear strain range and is the normal strain range on the plane experiencing the shear strain range . Considering elastic and plastic strains separately with the appropriate values of Poisson's ratio results in:

           (68)

Where,

A = 1.3+0.7S

B = 1.5+0.5S

Mean stress effects are included using Morrow's mean stress approach of subtracting the mean stress from the fatigue strength coefficient. The mean stress on the maximum shear strain amplitude plane, is one half of the axial mean stress leading to:

   (69)

Select either the Fatemi-Socie model or the Miller-Brown model for shear crack growth mode. The SWT model is always used for tensile crack growth.

If CHK is set to YES in the FATPARM Bulk Data Entry for proportional load cases, the above equation is calculated on the maximum shear strain plane. For more details, see Proportional Biaxial Load.

In case of proportional load, you do not have to search critical plane to find maximum damage plane, because direction of principal stress and principal strain do not change, you know the direction of plane where the maximum shear stress amplitude and the maximum normal stress amplitude. At the plane, you can easily calculate required information using Mohr's circle.

If CHK is set to YES in FATPARM for proportional load case, damage in Fatemi-Socie model is calculated on maximum shear strain plane, which is 45 degrees away from a principal stress plane.

Damage in a SWT model is calculated in the maximum principal stress plane.

Likewise, damages in Brown-Miller and Findley models are calculate on the maximum shear strain plane and maximum shear tress plane, respectively.

Figure 22: Analysis flow for strain-based multiaxial fatigue analysis

The Dang Van criterion is used to predict if a component will fail in its entire load history. In certain physical systems, components may be required to last infinitely long. For example, automobile components which undergo multiaxial cyclic loading at high rotational velocities (like propeller shafts) reach their high cycle fatigue limit within a short operating life. The conventional fatigue result that specifies the minimum fatigue cycles to failure is not applicable in such cases. It is not necessary to quantify the amount of fatigue damage, but just to consider if any fatigue damage will occur during the entire load history of the component. If damage does occur, the component cannot experience infinite life. Fatigue analysis based on the Dang Van criterion is designed for this purpose.

Fatigue crack initiation usually occurs at zones of stress concentration such as geometric discontinuities, fillets, notches and so on. This phenomenon takes place in the microscopic level and is localized to certain regions like grains which have undergone local plastic deformation in characteristic intra-crystalline bands. The Dang Van approach postulates a fatigue criterion using microscopic variables in the apparent stabilization state; this is a state of elastic shakedown if no damage occurs. The main principle of the criterion is that the usual characterization of the fatigue cycle is replaced by the local loading path and so damaging loads can be distinguished.

The general procedure of Dang Van fatigue analysis is:

1.	Evaluate the macroscopic stresses , for each location at a different point in time.

2.	Split the macroscopic stress into a hydrostatic part p(t) and a deviatoric part Sij(t).

3.	Calculate the stabilized microscopic residual stress based on the following equation:

        (70)

The expression is minimized with respect to and maximized with respect to t.

           (71)

          (72)

Where, b and a are material constants.

If FOS is less than 1, the component cannot experience infinite life.

OptiStruct Factor of Safety setup