PH8151 Engineering Physics Chapter 1

8 CHAPTER 1. PROPERTIES OF MATTER

The slope of the linear segment in Figure1.6corresponds to the modulus of elasticity

E. This modulus may be thought of as stiness, or a material’s resistance to elastic

deformation. The greater the modulus, the stier the material, or the smaller the elastic

strain that results from the application of a given stress.

Most materials follow Hooke’s law only at small strains (typically less than 1%). Metals obey Hooke’s

law with fully elastic behaviour only for very small strains (typically <0.2%). In this region, the extension

is usually both linear and recoverable. At larger strains, extension is non-linear and nonrecoverable.

Elastic Limit:The elastic limit (the point 2 in Figure1.6) is the limit beyond which

the material will no longer go back to its original shape when the load is removed.

The stress corresponding to this point is the maximum stress that may be developed

such that there is no permanent deformation when the load is entirely removed.

Elastic Range: The region in stress-strain diagram from 1 to 2 in figure1.6is called

the elastic range. The stress is linearly proportional to the strain in this region and

Hooke’s law (‡Ã‘) is obeyed.

Yield Point: As the material is deformed beyond the elastic limit (the point 2 Figure

1.6), the stress is no longer proportional to strain (Hooke’s law,‡=E‘ceases to

be valid), and permanent, nonrecoverable, or plastic deformation occurs. This is

called the thephenomenon of yielding. Yield strength is indicative of the stress at

which plastic deformation begins. The position of the point separating plastic and

elastic regions is dicult to measure precisely. As a consequence, a convention has

been established wherein a straight line is constructed parallel to the elastic portion

of the stress–strain curve at some specified strain oset, usually 0.002 as shown in

Figure1.7. The stress corresponding to the intersection (the point 3 Figure1.6)of

this line and the stress–strain curve as it bends over in the plastic region is defined

as the yield strength‡y.

intended. It is therefore desirable to know the stress level at which plastic defor-

mation begins, or where the phenomenon of yieldingoccurs. For metals that expe-

rience this gradual elastic–plastic transition, the point of yielding may be determined

as the initial departure from linearity of the stress–strain curve; this is sometimes

called the proportional limit,as indicated by point Pin Figure 6.10a,and represents

the onset of plastic deformation on a microscopic level. The position of this point

Pis difficult to measure precisely. As a consequence, a convention has been estab-

lished wherein a straight line is constructed parallel to the elastic portion of the

stress–strain curve at some specified strain offset, usually 0.002. The stress

corresponding to the intersection of this line and the stress–strain curve as it bends

over in the plastic region is defined as the yield strength.^8 This is demonstrated

in Figure 6.10a.Of course,the units of yield strength are MPa or psi.^9

For those materials having a nonlinear elastic region (Figure 6.6), use of the

strain offset method is not possible, and the usual practice is to define the yield

strength as the stress required to produce some amount of strain (e.g.,!! 0.005).

Some steels and other materials exhibit the tensile stress–strain behavior shown

in Figure 6.10b.The elastic–plastic transition is very well defined and occurs abruptly

in what is termed a yield point phenomenon.At the upper yield point,plastic de-

formation is initiated with an apparent decrease in engineering stress. Continued

deformation fluctuates slightly about some constant stress value, termed the lower

yield point; stress subsequently rises with increasing strain. For metals that display

this effect, the yield strength is taken as the average stress that is associated with

the lower yield point, because it is well defined and relatively insensitive to the

sy

6.6 Tensile Properties • 163

Stress

!y

!y

Stress

Strain Strain

ElasticPlastic

0.

P

Upper yield

point

Lower yield

point

(a) (b)

Figure 6.

(a) Typical stress–

strain behavior for

a metal showing

elastic and plastic

deformations, the

proportional limit P,

and the yield strength

as determined

using the 0.

strain offset method.

(b) Representative

stress–strain

behavior found for

some steels

demonstrating the

yield point

phenomenon.

sy,

(^8) Strengthis used in lieu of stressbecause strength is a property of the metal, whereas
stress is related to the magnitude of the applied load.
(^9) For customary U.S. units, the unit of kilopounds per square inch (ksi) is sometimes used
for the sake of convenience, where
1 ksi !1000 psi
yielding
proportional limit
yield strength
JWCL187_ch06_150-196.qxd 11/5/09 9:36 AM Page 163
Figure 1.7: Representative stress–strain behaviour for steels demonstrating the conven-
tion of determining yield stress.(Picture courtesy :[ 1 ])
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