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where
is Poisson’s ratio. Additionally, notice that
can also be measured
directly during the deformation experiment. The quantities
and
are
independent and completely determine isotropic linearly elastic stress-strain
behavior for isothermal loading.
Consider now an experiment where the sample is loaded along path OABC. The
following facts are pertinent:
• Above the stress state corresponding to point A, the relation between stress
and strain is no longer linear.
• Upon unloading from any point C above point A, the metal will not retrace the
original loading path, but rather it will unload along path CDE, which is
parallel to path OA. Unloading along path CDE occurs elastically.
• Once completely unloaded, the material will now possess a permanent,
inelastic strain (
= OE).
• If the bar at state E is now reloaded, the stress-strain behavior will be elastic
and follow path EDC. Continued loading beyond the stress level
corresponding to point C will follow path CFG.
• Point G represents the ultimate strength of the material. States beyond that
corresponding to point G are undefined.
3.6.5 Yield Stress
Behavior of the sample beyond point A is termed “inelastic.” Following the first
occasion for which the stress corresponding to point A is exceeded, it will be
necessary to compile a new set of constitutive (stress-strain) relations for the
material. Because of its importance as a limit point of elastic behavior
5
, the stress
corresponding to point A is given a special name, the initial yield stress, denoted
y
.
The yield stress,
y
, is a difficult, if not an impossible, number to obtain
experimentally. Close examination of the character of the curve in Figure 3-2 will
indicate that, experimentally, by the time one detects inelastic behavior, the yield
stress has already been exceeded. In order to circumvent this experimental
problem, it is common practice to approximate the yield stress by means of a
suitable alternate definition. For example, probably the most popular alternate
definition of yield stress is the so-called 0.2% offset stress, which is determined
by the intersection of the experimental stress-strain curve with a line parallel to
the elastic portion of the curve (i.e., slope =
) and passes through the point
σ
=
0,
ε
= 0.002 (such a line would be similar to line CDE in Figure 3-2). On the other
hand, API defines the yield stress as the stress corresponding to
ε
= 0.005
6
. As
5
For simplicity, the yield point, signifying the onset of inelastic behavior, and the
proportional limit, signifying the upper limit of applicability of Eq. (B-38) have been
assumed to coincide. However, for real materials, this is not always true.
6
The value of strain corresponding to yield may vary with grade. For example, the yield
stress for grade P-110 is defined as the stress corresponding to
= 0.006, and the yield
stress for grade Q-125 is defined as the stress corresponding to
= 0.0065.
3-10 Casing/Tubing Design Manual
October 2005