Chapter 2

MATERIALS AND PROCESSES

37

 

Tr

S

=

us

(2.5 a)

J

where T is the applied torque necessary to break the specimen, r is the radius of the 2

specimen, and J is the polar second moment of area of the cross section. The distribution of stress across the section loaded in torsion is not uniform. It is zero at the center and maximum at the outer radius. Thus the outer portions have already plastically yielded while the inner portions are still below the yield point. This nonuniform stress distribution in the torsion test (unlike the uniform distribution in the tension test) is the reason for calling the measured value at failure of a solid bar in torsion a modulus of rupture. A thin-walled tube is a better torsion-test specimen than a solid bar for this reason and can give a better measure of the ultimate shear strength.

In the absence of available data for the ultimate shear strength of a material, a reasonable approximation can be obtained from tension test data:*

steels :

S

us 0.80 Sut

(2.5 b)

other ductile metals :

S

us 0.75 Sut

Note that the shear yield strength has a different relationship to the tensile yield strength: S

ys 0.577 Sy

(2.5c)

+

This relationship is derived in Chapter 5, where failure of materials under static load-stress

ing is discussed in more detail.

 

time

 

Fatigue Strength and Endurance Limit

 

The tensile test and the torsion test both apply loads slowly and only once to the specimen. These are static tests and measure static strengths. While some machine parts may FIGURE 2- 9

see only static loads in their lifetime, most will see loads and stresses that vary with time.

 

Materials behave very differently in response to loads that come and go (called fatigue Time-Varying Loading

 

loads) than they do to loads that remain static. Most of machine design deals with the design of parts for time-varying loads, so we need to know the fatigue strength of materials under these loading conditions.

* In Chapter 14 on helical spring

design, an empirical relationship

One test for fatigue strength is the R. R. Moore rotating-beam test in which a similar, for the ultimate shear strength of

but slightly smaller, test specimen than that shown in Figure 2-1 is loaded as a beam in small diameter steel wire, based

on extensive testing of wire in

bending while being rotated by a motor. Recall from your first course in strength of torsion, is presented in equation

materials that a bending load causes tension on one side of a beam and compression on 14.4 (p. 793) and is Sus = 0.67

Sut. This is obviously different

the other. (See Sections 4-9 and 4-10 for a review of beams in bending.) The rotation than the general approximation

of the beam causes any one point on the surface to go from compression to tension to for steel in equation 2.5b. The

best data for material properties

compression each cycle. This creates a load-time curve as shown in Figure 2-9.

will always be obtained from tests

of the same material, geometry,

and loading as the part will be

The test is continued at a particular stress level until the part fractures, and the num-subjected to in service. In the

ber of cycles N is then noted. Many samples of the same material are tested at various absence of direct test data we

must rely on approximations of

stress levels S until a curve similar to Figure 2-10 is generated. This is called a Wohler the sort in equation 2.5b and

strength-life diagram or an S-N diagram. It depicts the breaking strength of a particu-apply suitable safety factors based

on the uncertainty of these

lar material at various numbers of repeated cycles of fully reversed stress.

approximations.

Image 78

 

38

MACHINE DESIGN -

An Integrated Approach

 

 

Sf

 

 

S

h

ut

failure line

2

tgn

 

ert se

 

 

An endurance limit S

u

e exists for some

gti

Se

Se

ferrous metals and titanium alloys.

fa

 

 

Other materials show no endurance limit.

g

 

Sf

lo

 

 

N

100 101 102 103 104 105 106 107 108 109

log number of cycles

F I G U R E 2 - 10

Wohler Strength-Life or S-N Diagram Plots Fatigue Strength Against Number of Fully Reversed Stress Cycles Note in Figure 2-10 that the fatigue strength Sf at one cycle is the same as the static strength Sut, and it decreases steadily with increasing numbers of cycles N (on a log-log plot) until reaching a plateau at about 106 cycles. This plateau in fatigue strength exists only for certain metals (notably steels and some titanium alloys) and is called the endurance limit Se. Fatigue strengths of other materials keep falling beyond that point.

While there is considerable variation among materials, their raw (or uncorrected) fatigue strengths at about N = 106 cycles tend to be no more than about 40–50% of their static tensile strength Sut. This is a significant reduction and, as we will learn in Chapter 6, further reductions in the fatigue strengths of materials will be necessary due to other factors such as surface finish and type of loading.

It is important at this stage to remember that the tensile stress-strain test does not tell the whole story and that a material’s static strength properties are seldom adequate by themselves to predict failure in a machine-design application. This topic of fatigue strength and endurance limit is so important and fundamental to machine design that we devote Chapter 6 exclusively to a study of fatigue failure.

The rotating-beam test is now being supplanted by axial-tension tests performed on modern test machines which can apply time-varying loads of any desired character to the axial-test specimen. This approach provides more testing flexibility and more accurate data because of the uniform stress distribution in the tensile specimen. The results are consistent with (but slightly lower-valued than) the historical rotating-beam test data for the same materials.

 

Impact Resistance

The stress-strain test is done at very low, controlled strain rates, allowing the material to accommodate itself to the changing load. If the load is suddenly applied, the energy absorption capacity of the material becomes important. The energy in the differential element is its strain energy density (strain energy per unit volume U 0), or the area under the stress-strain curve at any particular strain.

U =

0  d

(2.6a)

0

 

Image 79