Chapter 1

Concepts of Material Fatigue

1.1. Introduction

1.1.1. Reminders on the strength of materials

1.1.1.1. Hooke’s law

We accept that the strain at a point of a mechanical part is proportional to the elastic force acting on this point. This law assumes that the strains remain very small (elastic phase of the material). It enables us to establish a linear relationship between the forces and the deformation or between the stresses and the strains. In particular, if we consider the normal stress and the shear stress, we can write successively

[1.1]

[1.2]

where

E = Young’s modulus or elastic modulus

G = shear modulus or Coulomb’s modulus

εn = tensile strain parallel to the axis of the part (= ) if l is the initial length of the object and ∆l is its extension.

εt = relative strain in the plane of the cross-section

The following table gives several values of the Young’s modulus E:

Table 1.1. Some values of Young’s modulus

NOTE.– Hooke’s law is only an approximation of the real relationship between stress and strain, even for small stresses [FEL 59]. If Hooke’s law is perfectly respected, the stress strain process would thus be, below the elastic limit, thermodynamically reversible, with complete restitution of the energy stored in the material. Experience shows that this is not the case and that, even at very low levels of stress, a hysteresis exists. The process is never perfectly reversible.

1.1.1.2. Stress–strain curve

Engineering stress–strain curve

Let us consider the curve obtained by carrying out a tensile test on a cylindrical sample of length l, made of mild steel for example, and by tracing the traction force F according to the extension ∆l that the sample experiences or, which amounts to the same thing, the normal stress σ = according to the relative expansion ε = (strain). The test is carried out by making force F grow progressively, starting from zero.

The stress strain curve thus obtained, traced in the axes (σ, ε), has an identical shape since the changing of the variable corresponds to a proportional transformation (S cross-section, l useful length of the bar). This dimensionless diagram is characteristic of the material here, and not of the sample considered (Figure 1.1).

Figure 1.1. stress–strain diagram of a ductile material. σu = ultimate stress, σy = yield stress, σp = proportional limit, σF = fracture stress OA = linear region, AE = plastic region

This curve can be broken down into four arcs. Arc OA corresponds to the elastic region where the strain is reversible; the elongation there is proportional to the force (Hooke’s law):

[1.3]

(E = Young’s modulus, S0 = initial cross-section of the sample of length l).

This relation can also be written

[1.4]

(). In reality, the expansions ∆l of this zone are very small and the above curve is very badly proportioned.

There are several definitions of the elastic limit, chosen according to the case for an elongation of 0.01 %, 0.1 % or 0.2 %, the latter value being the most frequently used.

We call the proportional limit σp the maximum stress up to which the material does not show residual strain after unloading [FEO 69].

The BC zone, called the yielding region, corresponds to a significant stretching of the sample for an almost constant traction force. This stage has a variable length according to the materials; it can possibly be unnoticeable on some recordings. The strain is permanent and homogeneous.

The yield stress σy is the stress beyond which the strain increases without a notable increase in the load (point B). We call the ultimate tensile strength (UTS) σu the ratio between the maximum force Fmax that a sample can bear and the initial area S0 of the cross-section of the sample before testing (Figure 1.1):

[1.5]

The CD zone, strain hardening region, represents an elongation of the sample with the force which is produced much more slowly than in the elastic zone. Work hardening corresponds to a plastic strain of the metal at a temperature lower than the recrystallization temperature (which makes it possible to replace the strained, work-hardened structure with a new structure with reformed grains).

If, after having increased the force F from 0 to Fm such that the point m belongs to arc CD, the load is decreased, we notice that the point shows the straight segment mn going from m and parallel to 0A (Figure 1.2). For a zero load, there remains a residual elongation. This is called plastic extension. The strain is permanent.

Figure 1.2. Plastic expansion

Let us recall that as long as point (F, ∆l) remains on 0A, it describes this segment in the opposite direction if the load is taken back to zero. 0A is a perfectly elastic zone, not leading to a residual elongation.

If, from n, the sample is loaded again, the new diagram is made up of arcs nm, mDE (Figure 1.3). We note that the rectilinear segment (elastic zone) of the work-hardened bar is longer than 0A. A stretched material can thus bear greater loads without residual strain.

Figure 1.3. New diagram after plastic strain

The mechanical properties of a work-hardened metal are modified a lot: the elastic limit, the breaking load and the hardness are greatly increased, the expansion to fracture, the resistance and the necking are generally reduced.

It is in this zone that the neck is formed, the part of the sample where the crosssection reduces as quickly when the load increases, thus setting the future fracture area (necking phenomenon). The force F passes through a maximum (at D) when the relative reduction of the area S in this domain becomes equal to the relative increase of the stress.

Between D and E, the extension of the bar is produced with a reduction of the force F (the average stress in the area of the neck continues to grow however). Necking is when the specimen’s cross-section starts to stretch significantly. The size of the neck varies with the nature of the material.

When the metal begins to neck, as the cross-sectional area of the specimen decreases due to plastic flow, it causes a reversal of the engineering stress–strain curve; this is because the engineering stress is calculated assuming the original cross-sectional area (S0) before necking.

DE is the necking region [FEO69]. At E, the sample fractures. The fracture strength σF is the ratio between the load to fracture FF and the cross-sectional area S0:

[1.6]

These definitions assume that the cross-section and the length of the sample do not vary much during the application of the load. In most practical applications, this hypothesis leads to results that are precise enough. The stress–strain curve traced with these definitions is called the engineering stress–strain curve (Figure 1.4).

Figure 1.4. stress–strain diagram – ultimate tensile strength and true ultimate strength

True stress—strain curve

In reality, beyond the elastic limit, the dimensions of the sample change when the load is applied. It is thus more exact to define the stresses by dividing the applied force by the real cross-section of the sample.

We call the true tensile ultimate strength σut the ratio between the maximum force Fmax that a sample can bear and the area Sm t of the true cross-section of the sample when the force is equal to Fmax :

[1.7]

The true fracture strength σFt is the load at fracture FF divided by the true cross-sectional area SFt of the sample [LIU 69].

[1.8]

The stress–strain curve obtained in these conditions is called the true stress–strain curve (Figure 1.4).

Like the ultimate tensile strength, the true fracture strength can help an engineer to predict the behavior of the material, but is not itself a practical strength limit.

If S0 is the initial cross-sectional area of the piece and St is the area of the section after work hardening, we call the work-hardening rate the ratio

Finally, we call the strain at break δ (%) the average residual strain which takes place at the time of fracture, linked to a determined length of the sample. If d is the diameter of the bar before testing, the standard length chosen is 5d = l0

[1.9]

A material is more plastic the larger the value of δ . δ characterizes the ability of the material to show large residual strains without fracture.

The materials which, on the other hand, split without going...