Apart from nineteenth-century steam boilers, machines and equipment for high-temperature
operation have been developed principally in the 20th century. Energy conversion systems
based on steam turbines, gas turbines, high-performance automobile engines, and jet
engines provide the technological foundation for modern society.
All of these machines have in common the use of metallic materials at temperatures
where time-dependent deformation and fracture processes must be considered in their
design. The single valued time-invariant strain associated with elastic or plastic
design analysis in low-temperature applications is not applicable, nor is there in
most situations a unique value of fracture toughness that may be used as a limiting
condition for part failure. In addition to the phenomenological complexities of
time-dependent behavior, there is now convincing evidence that the synergism associated
with gaseous environmental interactions may have a major effect, in particular on
Basic Concepts of Elevated-Temperature Design. Time-dependent deformation and fracture
of structural materials at elevated temperatures are among the most challenging
engineering problems faced by materials engineers. In order to develop an improved
design methodology for machines and equipment operating at high temperatures, several
key concepts and their synergism must be understood:
- Plastic instability at elevated temperatures
- Deformation mechanisms and strain components associated with creep processes
- Stress and temperature dependence
- Fracture at elevated temperatures
- Environmental effects.
The issues of interest from a design basis are the nature of primary creep, the
validity of the concept of viscous steady-state creep, and the dependence of deformation
on both temperature and stress. The simplest and most pervasive idea in creep of metals
is an approach to an equilibrium microstructural and mechanical state. Thus a hardening
associated with dislocation generation and interaction is countered by a dynamic
microstructural recovery or softening. This process proceeds during primary creep and
culminates in a steady-state situation.
A major issue in the tensile creep test is the role of plastic instability in leading
to tertiary creep. Understanding of the nature of plastic instability for time-dependent
flow has depended on the theory of Hart. He showed that the condition for stable
γ + m ≥ 1
m is the strain-rate sensitivity, and
γ, is a measure of the strain-hardening rate.
For steady-stale flow, γ is equal to 0.
For constant stress tests, Burke and Nix concluded that flow must be unstable when
steady state is reached according to Hart’s criterion but that macroscopic
necking is insignificant and that the flow remains essentially homogeneous. They
concluded that a true steady state does exist. Hart himself questioned the conclusions
based on their analysis but did not rule out the possibility of a steady state for pure
In a very careful experimental analysis, Wray and Richmond later concluded that the
concept of a family of steady states is valid. Tests were performed in which two of
the basic parameters (stress, strain rate, and temperature) are held constant. However,
they reported the intrusion of nonuniform deformation before the steady state was
reached. They also pointed out the complexities associated with uncontrolled and often
unmeasured loading paths, which produce different structures at the beginning of the
constant stress or constant strain rate portions of the test. For constant stress
tests in pure metals, although the concept of steady state is appealing, it appears
not yet to have been rigorously demonstrated.
In constant load tests, steady-state behavior would of course result in an increasing
creep rate after the minimum, as the true stress increases. As such, the test is
inappropriate to evaluate the concept. However, it is by far the most common type of
creep test and can be analyzed for instability.
Creep behavior can be characterized either in terms of deformation mechanisms or in
terms of strain constituents.
Deformation Mechanisms. Creep of metals is primarily a result of
the motion of dislocations, but is distinct from time-independent behavior in that
flow continues as obstacles, which may be dislocation tangles or precipitate particles,
are progressively overcome. The rate-controlling step involves diffusion to allow
climb of edge dislocations or cross slip of screw dislocations around obstacles. In
steady-state theory, there is a balance between the hardening associated with this
dislocation motion and interaction, and a dynamic recovery associated with the
development of a dislocation substructure.
Theory for such a process predicts a power-law dependence of creep rate on applied
stress. At very high homologous temperatures (T/Tm) and low stresses, creep may
occur in both metals and ceramics by mass transport involving stress-directed flow of
atoms from regions under compression to regions under tension. In this case, theory i
ndicates that there is a stress dependence of unity and that the process is controlled
either by bulk diffusion or by grain-boundary diffusion. These various processes of
creep (dislocation controlled as well as diffusion controlled) may be represented
on a deformation mechanism map to highlight regimes of stress and temperature where
each mechanism, based on current theories, may be operating. However, such maps are
only as good as the theories on which they are based and give no guidance on deformation
Another important deformation process in metallic and ionic polycrystals at high
temperature and low stresses is grain-boundary sliding. The resistance to sliding is
determined by the mobility of grain-boundary dislocations and by the presence of hard
particles at the boundary. This sliding leads to stress concentrations at grain junctions,
which are important in nucleating cracks. In ductile materials, these stress
concentrations may be relieved by creep and stress relaxation in the matrix or by
Strain Components. There are several different sources of strain at
high temperature in response to an applied stress. The elastic strain is directly
proportional to stress, and a modulus that is temperature dependent can be determined.
For metallic materials and ceramics, although there is a strain-rate dependence of
elastic modulus, it is small and often ignored. Plastic strain for all materials may
be treated as three separate constituents:
- Time-independent nonrecoverable, which may be thought of as an instantaneous
- Time-dependent nonrecoverable, which may involve any or all of the micromechanisms
- Time-dependent recoverable.
The first of these is unlikely to be significant in practical applications except in
the region of stress concentrations since loading is normally well below the macroscopic
yield stress. The second is the major source of creep in normal laboratory testing.
The third constituent is not widely studied or analyzed, but may become very
important at low stresses and under nonsteady conditions, that is, high-temperature
service. It leads to what has been termed creep recovery and anelasticity.
At high temperatures, the application of a stress leads to creep deformation resulting
from the motion of dislocations, mass transport by diffusion, or grain-boundary sliding.
These processes in turn lead to a distribution of internal stresses that may relax on
removal of the stress. In metals it is associated with the unbowing of pinned dislocations,
rearrangement of dislocation networks, and local grain-boundary motion.
Whereas the importance of creep recovery is well recognized in polymer design, it has
often been ignored in design of metallic and ceramic materials. A few extensive studies
have been reported on metals that have led to several broad conclusions:
- Creep-recovery strain increases linearly with stress for a fixed time at a given
temperature, but is dependent on prestrain.
- The rate of creep recovery increases with increasing temperature.
- When the stress is low enough, essentially all transient creep is linear with
stress and recoverable.
- Mathematically, the recovery may be described by a spectrum of spring dashpot
combinations with a wide range of relaxation times.
Stress and Temperature Dependence
The minimum creep rate in both constant load and constant stress tests is normally
represented by a power function of stress, and the temperature by an Arrhenius e
xpression including an activation energy term (Q) derived from chemical reaction
where S, which is a constant, depends on structure. Although an exponential or
hyperbolic sine stress function may provide a better fit in some cases, the power
function has generally prevailed and has become strongly linked with mechanistic