s_m        – mean stress

            s_a        – stress amplitude

            Ds       – stress range

            s_max      – maximum stress

            s_min      – minimum stress

            R          – stress ratio:  

The cyclic stress can be fully characterized (apart from the frequency) by any combination of two of these parameters. The stress range, Ds, and the stress ratio, R, are the two most commonly used.  Note that in a constant-amplitude test each of these parameters has a constant value with respect to time.

 

Figure 5.1.2.  Definition of Terms for Fatigue Crack Growth and Stress Intensity

The stress history can be converted into a stress intensity factor history at a given crack length by multiplying the stress history by the stress intensity factor coefficient, as shown in Figure 5.1.2b.  The following parameters are defined:

            K_max     – maximum stress intensity factor 

            K_min      – minimum stress intensity factor 

            K_m        – mean stress intensity factor 

            K_a        – amplitude of the stress intensity factor 

            DK       – range of the stress intensity factor 

            R_K        – cycle ratio:

The above calculation schemes for stress intensity factor parameters, while being the most straightforward algebraically, have an operational quality about them.  For example, it is theoretically difficult to define a negative stress intensity factor that happens if the stress becomes compressive.  In this case, the crack closes and the crack tip stress field loses its singularity character; thus, the stress intensity factor ceases to have meaning.  The operational quality of the negative stress intensity factors calculated for compressive stress situations has been given a lot of consideration by the aerospace industry and by ASTM, specifically its subcommittee on sub-critical crack growth (ASTM E24.04).  ASTM has chosen to provide the following definitions when the minimum stress (s_min) is less than zero:

Kmin = 0 if smin < 0

DK = Kmax if smin < 0

The reader should be aware of the ASTM definition of DK because that convention is used in the Damage Tolerant Design (Data) Handbook [1994] for the presentation of crack growth rate data when part of the fatigue cycle is compressive, i.e., when s_min < 0 (R < 0).  The algebraic definition of DK is used in the current version of MIL-HDBK-5.  Before negative stress ratio (R < 0) data are used, it is important to establish the operational definition of DK.  The reader should note that the behavior of the material under negative stress ratio conditions is itself independent of the operational definition of DK.

In the elastic case, the stress-intensity factor alone is sufficient to describe the stress field at the tip of a crack.  When the plastic zone at the crack tip is small compared with the crack size, the stress-intensity factor gives a good indication of the stress environment of the crack tip.  Two different cracks that have the same stress environment (equal stress-intensity factors) will behave in the same manner and show the same rate of growth.

Since two parameters are required to characterize the fatigue cycle, two parameters are required to characterize crack growth rate behavior.  The crack growth rate per cycle, da/dN, can be generally described with functional relation of the type:

(5.1.1)

where a is the crack length, N is the number of cycles, and R is the stress ratio associated with the stress cycle.

 

 

Typically, fatigue crack growth rate data is described using plots of da/dN versus DK on double-logarithmic scale graph paper.  Figure 5.1.3 presents fatigue crack growth rate data for 7075 aluminum in the graphical format that is used in the Damage Tolerant Design (Data) Handbook [1994].  Figures 5.1.4 and 5.1.5 describe example composite da/dN data plots for 7075 aluminum as a function of DK (algebraic definition) for different stress ratio (R) values [MIL-HDBK-5H, 1998].  Both Figures 5.1.4 and 5.1.5 provide mean trend curves that represent the function f(DK, R) in Equation 5.1.1.  On the basis of these figures, it can be seen that f(DK, R) is not a simple function.  Figure 5.1.6 is a schematic illustration of fatigue crack growth rate behavior from the threshold region (below 10^-8 inch/cycle) to the onset of rapid cracking in the fracture toughness region (above 10^-3 inch/cycle).  As can be seen from Figures 5.1.3 - 5.1.6, the behavior exhibits a sigmoidial shape suggesting that there might be asymptotes at the two extreme regions.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1.3.  Fatigue Crack Growth Rate Data Presentation Format Used in the Damage Tolerant Design (Data) Handbook [1994].  Data Presented for Two Stress Ratios for 7057-T7351 Aluminum Alloy

Figure 5.1.4.  Sample Fatigue Crack Growth Rate Data for 7075-T6 Aluminum Alloy Sheet From MIL-HDBK-5H [1998]

Figure 5.1.5.  Sample Fatigue Crack Growth Rate Data for 7075-T7351 Aluminum Alloy Plate From MIL-HDBK-5H [1998]

Figure 5.1.6.  Schematic of Fatigue Crack Growth Rate Behavior