Page 193 - Academic Press Encyclopedia of Physical Science and Technology 3rd Polymer

P1: GLM Final Pages
Encyclopedia of Physical Science and Technology EN012c-598 July 26, 2001 15:59

702 Polymers, Mechanical Behavior

due to the thin bond line, considerable shear strain may Integration of Eq. (6) leads to
well have been imposed.
ε t = ln(l /l 0 ). (7)
Returning to tensile deformation, the principal strain
or deformation direction is correlated to another common One immediately notes that ε t (also known as the Henky
parameter that is often used to express the level of defor- strain) is also given by the natural logarithm of the re-
mation, that is, the percent elongation: spective extension ratio for the corresponding axis. In the
deformation behavior of polymeric systems, the true strain
% elongation = ε × 100. (2)
variable is not often utilized and it is more common for
A third means of expressing deformation is to use the one of the other three parameters to be quoted. However,
extension ratios or draw ratios, which we shall denote by some data do exist in the literature in the form of true
λ i . Three such values exist, one correlated to each of the strain; this variable is quite different from the others due
principal axes. These are defined as to its logarithmic nature.
λ x = l x /l 0x ,

λ y = l y /l 0y , (3) III. IMPORTANT STRESS–STRAIN
DEFORMATION PARAMETERS
λ z = l z /l 0z .
In the limit of zero deformation, each λ i takes on the value The mode of tensile deformation will be used to elu-
of unity, in contrast to 0 for the respective values of ε i . cidate some of the important parameters describing the
Hence, there is a numerical factor of unity that relates λ i mechanical properties of a material. Figure 5 is a gen-
to ε i , that is, eral sketch of a common stress–strain response exhib-
λ i = ε i + 1. (4) ited by many polymeric materials—particularly ductile
semicrystalline materials like polyethylene or polypropy-
There are certain relationships between the λ terms that lene. What can one extract from these data that might
are useful with regard to noting how the dimensions of help convey the mechanical response of the system and
a specimen change in a given deformation. In particular, that may be useful in deciding on the applicability of
for materials that undergo a constant-volume deforma- a given material? Often, the values of σ b and ε b (see
tion, the product λ x λ y λ z is unity. In fact, good elastomers Fig. 5) will be of importance since these are related to
nearly follow this behavior, and since they deform uni- the stress at break and to the strain at break, the latter of
formly (homogeneously), it can be shown that for uniaxial which can now easily be converted to percent elongation,
deformation along z
the extension ratio, or true strain at break. The values σ y
λ x = λ y = λ −1/2 . (5) and ε y are particularly important and should be distinctly
z
noted if this “peak” occurs in the stress–strain response.
The same approximation can often be used to define the The respective stress and strain correlated with this peak
deformation of many other materials, although some error
is introduced, as pointed out in Section III.
Although the derivation will not be carried out here, it
can be shown that if a constant volume homogeneous ten-
sile deformation occurs be it uniaxial, biaxial, etc., then
utilizing the fact that the product of the three extension
ratios is unity, it follows that the true stress in a given
principal deformation direction is equal to the engineering
stress in that same direction multiplied by the respective
extension ratio along that same axis. This can be a useful
relationship to interconvert between true and engineering
stress when the assumption of constant volume deforma-
tion is well approximated.
A final parameter describing the degree of deformation
is the true strain ε t . We establish this parameter by utilizing
a differential form of our earlier definition of strain above,
that is,
FIGURE 5 Generalized stress–strain curve that shows distinct
d ε i = dl i /l i . (6) yield.