# Fatigue Analysis Of Structural Materials Construction Essay

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## CHAPTER 3

## 3.1 INTRODUCTION

In previous chapter, concept of Tensile Test, elastic constant, surface energy and fracture toughness along with derivation of fatigue potential energy, lethargy coefficient, surface energy and fracture toughness are discussed. Fatigue is a phenomenon associated with variable loading or more precisely to cyclic stressing or straining of a material. Just as the human beings get fatigue when a specific task is repeatedly performed, in a similar manner metallic components subjected to variable loading get fatigue, which leads to their premature failure under specific conditions. Fatigue loading is primarily the type of loading which causes cyclic variations in the applied stress or strain on a component. Thus any variable loading is basically a fatigue loading.

Fatigue is the behavior of materials under fluctuating and reversing loads. This behavior of material is different from the behavior under the study load. The rate of loading is usually not a factor in fatigue behavior. Fatigue behavior is experienced by all materials whether metals, plastics, concrete, or composites. The main effects of fatigue on the properties of materials are

Loss of ductility.

Loss of strength.

Enhanced uncertainty in strength and the service life of materials [15].

## 3.2 FATIGUE PROPERTIES

## (a). Maximum stress

The maximum value of stress in the stress cycle is known as maximum stress. It is denoted by .

## (b). Range of stress

The range of stress is difference of maximum stress and minimum stress. It is denoted by

## (c). Mean stress

The mean stress is average value of maximum stress and minimum stress. It is denoted by

## (d). Stress ratio

The stress ratio is defined as ratio of minimum stress and maximum stress. It is denoted by ()

R =

Fig 3.1: Fluctuating tensile stress [15]

## (e). Fatigue life

The fatigue life is defined as the number of stress cycle which can be sustained without fracture.

## (f). Cycle ratio

Cycle ratio is defined as the ratio of number of cycle applied and fatigue life.

## (g). Fatigue life

The fatigue life is defined as the limiting value of stress below which material can endure infinite number of cycle of stress.

## (h). Fatigue strength

The fatigue strength is defined as the greatest stress which can be sustained for a given number of cycles without fracture.

## (i). Fatigue ratio

The fatigue ratio is defined as the ratio of fatigue limit and tensile strength [15].

## 3.3 STRESSES IN FATIGUE FAILURE

## (a). Alternating/Reversing Stress

Fig 3.2: Alternating/Reversing Stress [16].

## (b). Repeated Stress

Fig 3.3: Repeated Stress [16]

## (c).Combined steady and cyclic stress

Fig 3.4: Combined steady and cyclic stress [16]

## 3.4 Cyclic Loading

There are essentially two types of fatigue loading:

Constant amplitude, proportional loading

Constant amplitude, non-proportional loading

## 3.4.1 Constant Amplitude, Proportional Loading

The constant amplitude loading (fully reversed loading) as shown in Fig 3.5. The loading ratio is defined as the ratio of the second load to the first load (LR = L2/L1).

Amplitude

Figure 3.5: Constant amplitude fully reversed loading. [17]

## 3.4.2 Constant Amplitude, Non-Proportional Loading

Constant amplitude non-proportional loading is shown in Fig. 3.6. The loading is of constant amplitude but non-proportional since principal stress or strain axes are free to change between the two load sets. No cycle counting is required. The critical fatigue location may occur at a spatial location that is not easily identifiable by looking at either of the base loading stress states.

Amplitude

Figure 3.6: Constant amplitude, non-proportional loading [17].

## 3.5 Mechanism OF Fatigue Failure

A fatigue failure begins with a small crack; the initial crack may be so minute and cannot be detected by the eyes or ordinary touch. The crack usually develops at a point of localized stress concentration like discontinuity in the material, such as a change in cross section, a keyway or a hole. Once a crack is initiated, the stress concentration effect becomes more pronounced and the crack propagates. Consequently the stressed area decreases in size, the stress increase in magnitude and the crack propagates more rapidly. Until finally, the remaining area is unable to sustain the load and the component fails/breaks suddenly. Thus, fatigue loading results in sudden, unwarned failure. The three modes of cracking used in fracture mechanics is shown in Fig. 3.7.

Figure 3.7: Three modes of cracking in fracture mechanics [18]

## 3.5.1 Crack Initiation

The areas of localized stress concentrations such as fillets, notches, key ways, bolt holes and even scratches or tool marks are potential zones for crack initiation. Crack also generally originates from a geometrical discontinuity or metallurgical stress raiser like sites of inclusions. As a result of the local stress concentrations at these locations, the induced stress goes above the yield strength (in normal ductile materials) and cyclic plastic straining results due to cyclic variations in the stresses.

Figure 3.8: The transition form of crack growth [18]

## 3.5.2 Crack Propagation

The crack further increases the stress levels and the process continues, propagating the cracks across the grains or along the grain boundaries, slowly increasing the crack size. As the size of the crack increases the cross sectional area resisting the applied stress decreases and reaches a threshold level at which it is insufficient to resist the applied stress. Fig. 3.8 shows the transition form of crack growth.

## 3.5.3 Fatigue Fracture

As the area becomes too insufficient to resist the induced stresses, any further increase in stress level causes a sudden fracture of the component.

## 3.6 Stress-Life CURVE

Several methods are available for the design of components subjected to fatigue loading. These methods require similar type of information such as identification of component locations for failure, the load spectrum for the structure or component, the stresses or strains at the candidate locations resulting from the loads, the temperature, the corrosive environment, the material behavior, and a methodology that combines all these effects to give a life prediction. The life of the components can be predicted using the followings:

stress vs number of cycle curves

hot-spot stresses

strain life, and

fracture mechanics.

With the exception of hot-spot stress method, all the procedures have been used for the design of aluminum structures. Since the well-known work of Wohler in Germany started in 1850's, engineers have employed curves of stress versus cycles to fatigue failure, which are often called S-N curves (stress-number of cycles) or Wohler's curve [18].

The basis of the stress-life method is the Wohler S-N curve, It is a plot of alternating stress 'S', versus number of cycles to failure 'N'. The data which results from these tests can be plotted on a curve of stress versus number of cycles to failure. This curve shows the scatter of the data taken for this simplest of fatigue tests. Fig. 3.9 shows a typical S-N material data in which arrows imply that the specimen had not failed in 107 cycles.

Figure 3.9 A typical S-N material data [18]

The stress based approach is widely used for the design of aluminum structure. Comparing the stress-time history at the chosen critical point with the S-N curve allows a life estimate for the component to be made. Stress-life approach assumes that all stresses in the component, is below the elastic limit at all times. It is suitable when the applied stress is nominally within the elastic range of the material and the number of cycles to failure is large. In high cycle fatigue the nominal stress approach is best suited. High cycle fatigue is one of the two regimes of fatigue phenomenon that is generally considered for metals and alloys. It involves nominally linear elastic behavior and causes failure after more than about 104 to 105 cycles. This regime is associated with lower loads and long lives, or high number of cycles to produce fatigue failure. As the loading amplitude decreases, the cycles-to-failure increases [18].

Fig 3.10: S-N curve for 1045 steel and 2014-T6 Aluminium [19]

## 3.7 GENERATION OF S-N CURVE FROM TENSILE TEST

For high-cycle loading, plastic micro strains are accumulated in the persistent slip bands, band strain-hardening as microstructure accumulate. The strain-hardening leads to crack formation, and seems to be related to formation of intrusions within bands. Crack propagation gives rise to fatigue fracture. Therefore the S-N curve at a particular stress level represents the formation of persistent slip band, its strain-hardening and crack formation (necking), and step-wise crack propagation locally in the test specimen. However, in the Tensile Test the whole specimen deforms plastically and strain-hardens homogeneously as the stress increases. At the final stage of strain-hardening, local necking occurs and proceeds to failure [12]. However accumulations of strain per unit volume in the deforming region before failure in both cases are equal due to same materials. It can be done graphically, shown in figure 3.11 (a) and (b).

Fig 3.11: (a) S-N curve [12]

Fig 3.11: (b) tensile test [12]

in the S-N curve and strain in tensile test at each stress level have a linear relationship with each other.

figure 3.12 (a) and (b) shows the S-N curves for mild steel and 4340 steel

Fig 3.12: S-N curve for common structural metals (a) mild steel (b) 4340 steel [12].

figure 3.13 (a), (b) and (c) shows the tensile-Test curves of mild steel at three stress rates

Fig 3.13: Tensile Test for mild steel at: (a) (b) (c) [12]

Fig 3.14: Tensile test for medium steel at: (a) (b) [12]

and and the elongations and have a linear relationship, = 24:as shown in Figs 3.15 and 3.16

log N

8

6

0.3

0.2

4

2

0.1

Fig 3.15: vs elongation for mild steel [12]

log N

8

6

0.3

0.2

4

2

0.1

Fig 3.16: vs elongation for medium carbon steel [12]

Both mild steel and medium-carbon steel have the same linear relationship. The number of cycles to failure depends only upon the elongation accumulated in the test and Young's modulus of material. The equations for both mild steel and medium carbon steel are the same, is shown in figure 3.17

log N

8

6

0.3

0.2

4

2

0.1

Fig 3.17: vs elongation for structural carbon steel [12]

The number of cycles to failure depends only upon elongation accumulated in the test and Young's modulus of the material. The fatigue life fraction passed in the test is given by

(3.4)

Integration gives

=1+eUo/kT (3.5) Eq (3.5) is approximated during plastic deformation as

(3.6)

Eq. (3.6) is simplified as

γ = kT (3.7)

γ =kT= (3.8)

From eq (3.8) a very important equation is acquired

ln Ny= (3.9)

If the test velocity is constant,

ln Ny=

Finally,

ln Ny

log Ny (3.10)

Eq.(3.10) explains the results of fig 3.17. If Eq.(3.10) is rewritten,we have

log Ny=Y (3.11)

In high-cycle fatigue test, greater elastic strain results into greater plastic microstrain. The magnitude of elastic strain is a function of Young's modulus is given by Hook's law. The magnitude of Young's modulus is divided into 4 groups: carbon steel; stainless steel; aluminium alloys and copper alloys. The value of Y in Eq. (3.11) can be written as [13].

Y = (-0.3)(E/106) + 33 (3.12)

Here, E has units of psi (1 psi = 0.703 g/mm2). The resultant Y values are 24 for structural carbon steel and 30 for aluminum alloys as shown in the figure 3.18 and 3.19.

log N

8

6

0.3

0.2

4

2

0.1

Fig 3.18: Relationship between log and for structural aluminium [12]

figure 3.19 shows the S-N curve for two structural metals (a) 7075 T-6 Aluminium and (b) 2024 T-3 Aluminium Fig 3.19: S-N curve for two common structural metals: (a) 7075 T-6 aluminium; (b) 2024 T-3 aluminium [12]

## 3.8 CONCLUDING REMARKS

In this chapter, concept of fatigue analysis of structural materials, different stresses in fatigue, cyclic loading and mechanism of fatigue failure are presented. The number of cycles to failure depends only upon the elongation accumulated in tensile test and Young's modulus of materials which is the main concept of generation of S-N curve from tensile -test.

In the next chapter, the values of lethargy coefficient, fatigue potential energy, surface energy and fracture toughness for different structural materials has been calculated and verified with ASTM standards.