Deformation - Orthopaedic Biomaterials in Research and Practice - page 31

Table 2.3 Conversion of load and deformation of a bar into stress and strain

Stress a (MPa) Strain

State Load (N) w (cm) w 2 (cm 2 ) Nominal True L (cm) Nominal True

None 1.0 1.0 0.0 0.0 4.0 0.0 0.0

a 200 0.98 0.96 200 210 4.2 0.05 0.049

b 400 0.96 0.92 400 435 4.4 0.1 0.095

c 700 0.92 0.85 700 830 4.8 0.2 0.182

d 700 0.75 0.56 700 1245 5.2 0.3 0.58

e 300 0.6 0.36 300 835 5.6 0.4 1.02

Rounded off to the nearest 5 MPa.

a

These definitions are for quantities called nominal or engineering
strain and stress. Nominal values are easy to calculate and are correct
for small elastic deformations. True values may be defined, which are
valid over wide ranges of both elastic and plastic deformation.

True stress and
strain

True values for stress and strain can be thought as accounting for the
instantaneously changing length and cross-sectional area, respectively.
True strain is defined as:

ε = ln( L / L 0 ) (before onset of necking)

= ln( A 0 / A )* (after onset of necking)

where the value of cross-sectional area ( A ) that corresponds to a particular
stress beyond that which produces necking is measured in the neck region
and is determined after loading to that stress followed by load release.
True stress is defined as:

σ = F / A

where A is the instantaneous cross-sectional area (in the neck region, if
necking has occurred) rather than the original cross-sectional area.
Let us take the data (Table 2.1) we obtained in the experiment of
stretching a bar (Figure 2.1) and convert the load and deformation values
into stress and strain (Table 2.3).
For the sets of values below the onset of necking (between c and d ),
there is a very good correspondence between nominal and true values of
both stress and strain. Because of this, and the fact that most devices are
designed for very small peak strains, nominal and true values tend to be
used interchangeably and the Greek letters sigma (σ) and epsilon (ε) are
used conventionally for all values of stress and strain.
Plotting the true values of stress as a function of true strain and con-
necting them with a smoothly curving line produces a stress-strain
curve (Figure 2.9). This expresses the intrinsic behavior of the material

* This form is correct; the apparent inversion from the prenecking formula occurs since
volume (= L 0 A 0 ) is conserved. “ln(x)” is the natural logarithm of x .